**Preface to "Theoretical Computer Science and Discrete Mathematics"**

This book includes 15 articles published in the Special Issue "lTheoretical Computer Science and Discrete Mathematics" of *Symmetry* (ISSN 2073-8994). This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim was to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The Special Issue covers topics in discrete mathematics including (but not limited to) graph theory, cryptography, numerical semigroups, discrete optimization, algorithms, and complexity.

The response to our call for papers had the following statistics:


We found the edition and selections of papers for this book very inspiring and rewarding. We thank the editorial staff and reviewers for their great efforts and help during the process.

## **Juan Alberto Rodr´ıguez Vel´azquez, Alejandro Estrada-Moreno**

*Editors*

## *Article* **Secure** *w***-Domination in Graphs**

## **Abel Cabrera Martínez , Alejandro Estrada-Moreno \* and Juan A. Rodríguez-Velázquez**

Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain; abel.cabrera@urv.cat (A.C.M.); juanalberto.rodriguez@urv.cat (J.A.R.-V.)

**\*** Correspondence: alejandro.estrada@urv.cat

Received: 31 October 2020; Accepted: 23 November 2020; Published: 25 November 2020

**Abstract:** This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure *w*-domination in graphs, where *w* = (*w*0, *w*1, . . . , *w<sup>l</sup>* ) is a vector of nonnegative integers such that *w*<sup>0</sup> ≥ 1. The secure *w*-domination number is defined as follows. Let *G* be a graph and *N*(*v*) the open neighborhood of *v* ∈ *V*(*G*). We say that a function *f* : *V*(*G*) −→ {0, 1, . . . , *<sup>l</sup>*} is a *<sup>w</sup>*-dominating function if *<sup>f</sup>*(*N*(*v*)) = <sup>∑</sup>*u*∈*N*(*v*) *f*(*u*) ≥ *w<sup>i</sup>* for every vertex *v* with *<sup>f</sup>*(*v*) = *<sup>i</sup>*. The weight of *<sup>f</sup>* is defined to be *<sup>ω</sup>*(*f*) = <sup>∑</sup>*v*∈*V*(*G*) *f*(*v*). Given a *w*-dominating function *f* and any pair of adjacent vertices *v*, *u* ∈ *V*(*G*) with *f*(*v*) = 0 and *f*(*u*) > 0, the function *fu*→*<sup>v</sup>* is defined by *fu*→*v*(*v*) = 1, *fu*→*v*(*u*) = *f*(*u*) − 1 and *fu*→*v*(*x*) = *f*(*x*) for every *x* ∈ *V*(*G*) \ {*u*, *v*}. We say that a *w*-dominating function *f* is a secure *w*-dominating function if for every *v* with *f*(*v*) = 0, there exists *u* ∈ *N*(*v*) such that *f*(*u*) > 0 and *fu*→*<sup>v</sup>* is a *w*-dominating function as well. The secure *w*-domination number of *G*, denoted by *γ s <sup>w</sup>*(*G*), is the minimum weight among all secure *w*-dominating functions. This paper provides fundamental results on *γ s <sup>w</sup>*(*G*) and raises the challenge of conducting a detailed study of the topic.

**Keywords:** secure domination; secure Italian domination; weak roman domination; *w*-domination

## **1. Introduction**

Let Z <sup>+</sup> <sup>=</sup> {1, 2, 3, . . . } and <sup>N</sup> <sup>=</sup> <sup>Z</sup> <sup>+</sup> ∪ {0} be the sets of positive and nonnegative integers, respectively. Let *G* be a graph, *l* ∈ Z <sup>+</sup> and *<sup>f</sup>* : *<sup>V</sup>*(*G*) −→ {0, . . . , *<sup>l</sup>*} a function. Let *<sup>V</sup><sup>i</sup>* <sup>=</sup> {*<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*) : *f*(*v*) = *i*} for every *i* ∈ {0, . . . , *l*}. We identify *f* with the subsets *V*0, . . . , *V<sup>l</sup>* associated with it, and thus we use the unified notation *f*(*V*0, . . . , *V<sup>l</sup>* ) for the function and these associated subsets. The weight of *f* is defined to be

$$
\omega(f) = f(V(\mathcal{G})) = \sum\_{i=1}^{l} i|V\_i|.
$$

Let *w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup>* such that *w*<sup>0</sup> ≥ 1. As defined in [1], a function *f*(*V*0, . . . , *V<sup>l</sup>* ) is a *w*-*dominating function* if *f*(*N*(*v*)) ≥ *w<sup>i</sup>* for every *v* ∈ *V<sup>i</sup>* . The *w*-*domination number* of *G*, denoted by *γw*(*G*), is the minimum weight among all *w*-dominating functions. For simplicity, a *w*-dominating function *f* of weight *ω*(*f*) = *γw*(*G*) is called a *γw*(*G*)-function. For fundamental results on the *w*-domination number of a graph, we refer the interested readers to the paper by Cabrera et al. [1], where the theory of *w*-domination in graphs is introduced.

The definition of *w*-domination number encompasses the definition of several well-known domination parameters and introduces new ones. For instance, we highlight the following particular cases of known domination parameters that we define here in terms of *w*-domination: the domination number *γ*(*G*) = *γ*(1,0) (*G*) = *γ*(1,0,...,0) (*G*), the total domination number *γt*(*G*) = *γ*(1,1) (*G*) = *γ*(1,...,1) (*G*), the *k*-domination number *γ<sup>k</sup>* (*G*) = *γ*(*k*,0) (*G*), the *<sup>k</sup>*-tuple domination number *<sup>γ</sup>*×*<sup>k</sup>* (*G*) = *γ*(*k*,*k*−1) (*G*), the *<sup>k</sup>*-tuple total domination number *<sup>γ</sup>*×*k*,*<sup>t</sup>* (*G*) = *γ*(*k*,*k*) (*G*), the Italian domination number *γI*(*G*) = *γ*(2,0,0) (*G*), the total Italian domination number *γtI*(*G*) = *γ*(2,1,1) (*G*), and the {*k*}-domination number *<sup>γ</sup>*{*k*} (*G*) = *<sup>γ</sup>*(*k*,*k*−1,...,0) (*G*). In these definitions, the appropriate restrictions on the minimum degree of *G* are assumed, when needed.

For any function *f*(*V*0, . . . , *V<sup>l</sup>* ) and any pair of adjacent vertices *v* ∈ *V*<sup>0</sup> and *u* ∈ *V*(*G*) \ *V*0, the function *fu*→*<sup>v</sup>* is defined by *fu*→*v*(*v*) = 1, *fu*→*v*(*u*) = *f*(*u*) − 1 and *fu*→*v*(*x*) = *f*(*x*) whenever *x* ∈ *V*(*G*) \ {*u*, *v*}.

We say that a *w*-dominating function *f*(*V*0, . . . , *V<sup>l</sup>* ) is a *secure w*-*dominating function* if for every *v* ∈ *V*<sup>0</sup> there exists *u* ∈ *N*(*v*) \ *V*<sup>0</sup> such that *fu*→*<sup>v</sup>* is a *w*-dominating function as well. The *secure w*-*domination number* of *G*, denoted by *γ s <sup>w</sup>*(*G*), is the minimum weight among all secure *w*-dominating functions. For simplicity, a secure *w*-dominating function *f* of weight *ω*(*f*) = *γ s <sup>w</sup>*(*G*) is called a *γ s <sup>w</sup>*(*G*)-function. This approach to the theory of secure domination covers the different versions of secure domination known so far. For instance, we emphasize the following cases of known parameters that we define here in terms of secure *w*-domination.


For the graphs shown in Figure 1, we have the following:

• *γ s* (1,1) (*G*1) = *γ s* (2,0) (*G*1) = *γ s* (2,1) (*G*1) = *γ*(2,0) (*G*1) = *γ*(2,1)*G*1) = *γ s* (1,1,0) (*G*1) = *γ s* (1,1,1) (*G*1) = *γ s* (2,0,0) (*G*1) = *γ s* (2,1,0) (*G*1) = *γ*(2,0,0) (*G*1) = *γ*(2,1,0) (*G*1) = *γ*(2,2,0) (*G*1) = *γ*(2,2,1) (*G*1) = *γ*(2,2,2) (*G*1) = 4 and *γ s* (2,2) (*G*1) = *γ*(2,2) (*G*1) = *γ s* (2,2,0) (*G*1) = *γ s* (2,2,1) (*G*1) = *γ s* (2,2,2) (*G*1) = *γ s* (3,0,0) (*G*1) = *γ s* (3,1,0) (*G*1) = *γ s* (3,1,1) (*G*1) = *γ s* (3,2,0) (*G*1) = *γ s* (3,2,1) (*G*1) = *γ s* (3,2,2) (*G*1) = *γ*(3,0,0) (*G*1) = *γ*(3,1,0) (*G*1) = *γ*(3,1,1) (*G*1) = *γ*(3,2,0) (*G*1) = *γ*(3,2,1) (*G*1) = *γ*(3,2,2) (*G*1) = 6.

$$\bullet \quad \gamma\_{(1,1)}^s(G\_2) = \gamma\_{(1,1,0)}^s(G\_2) = \gamma\_{(1,1,1)}^s(G\_2) = \gamma\_{(2,2,0)}(G\_2) = \gamma\_{(2,2,1)}(G\_2) = \gamma\_{(2,2,2)}(G\_2) = \mathfrak{3}.$$

• *γ s* (1,1) (*G*3) = *γ s* (1,1,0) (*G*3) = *γ s* (1,1,1) (*G*3) = *γ*(2,1,0) (*G*3) = *γ*(3,0,0) (*G*3) = 3 < 4 = *γ s* (2,0,0) (*G*3) = *γ s* (2,1,0) (*G*3) = *γ s* (3,1,0) (*G*3) = *γ*(2,2,0) (*G*3) = *γ*(2,2,1) (*G*3) = *γ*(2,2,2) (*G*3) = *γ*(3,2,0) (*G*3) < 5 = *γ s* (2,2,0) (*G*3) = *γ s* (3,2,0) (*G*3) = *γ s* (2,2,1) (*G*3) = *γ s* (2,2,2) (*G*3) = *γ s* (3,1,1) (*G*3) = *γ s* (3,2,1) (*G*3) = *γ*(3,2,1) (*G*3) = *γ*(3,2,2) (*G*3) < 6 = *γ s* (3,2,2) (*G*3).

This paper is devoted to providing general results on secure *w*-domination. We assume that the reader is familiar with the basic concepts, notation, and terminology of domination in graph. If this is not the case, we suggest the textbooks [20,21]. For the remainder of the paper, definitions are introduced whenever a concept is needed.

**Figure 1.** The labels of black-colored vertices describe the positive weights of a *γ s* (2,1,0) (*G*1)-function, a *γ s* (1,1,1) (*G*2)-function, and a *γ s* (2,2,2) (*G*3)-function, respectively.

#### **2. General Results on Secure** *w***-Domination**

Given a *w*-dominating function *f*(*V*0, . . . , *V<sup>l</sup>* ), we introduce the following notation.


Obviously, if *f* is a secure *w*-dominating function, then *M<sup>f</sup>* (*v*) 6= ∅ for every *v* ∈ *V*0.

**Lemma 1.** *Let f be a secure w-dominating function on a graph G, and let u* ∈ M*<sup>f</sup>* (*G*)*. If T<sup>f</sup>* (*u*) 6= ∅*, then each vertex belonging to T<sup>f</sup>* (*u*) *is adjacent to every vertex in D<sup>f</sup>* (*u*) *and, in particular, G*[*T<sup>f</sup>* (*u*)] *is a clique.*

**Proof.** Since *T<sup>f</sup>* (*u*) ⊆ *D<sup>f</sup>* (*u*), we only need to suppose the existence of two non-adjacent vertices *v* ∈ *T<sup>f</sup>* (*u*) and *v* <sup>0</sup> ∈ *D<sup>f</sup>* (*u*) with *v* 6= *v* 0 . In such a case, *<sup>f</sup>u*→*<sup>v</sup>* <sup>0</sup>(*N*(*v*)) < *w*0, which is a contradiction. Therefore, the result follows.

**Remark 1** ([1])**.** *Let G be a graph of minimum degree δ and let w* = (*w*0, *w*1, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> . If w*<sup>0</sup> ≥ *w*<sup>1</sup> ≥ · · · ≥ *w<sup>l</sup> , then there exists a w-dominating function on G if and only if w<sup>l</sup>* ≤ *lδ*.

Throughout this section, we repeatedly apply, without explicit mention, the following necessary and sufficient condition for the existence of a secure *w*-dominating function on *G*.

**Remark 2.** *Let G be a graph of minimum degree δ and let w* = (*w*0, *w*1, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> . If w*<sup>0</sup> ≥ *w*<sup>1</sup> ≥ · · · ≥ *w<sup>l</sup> , then there exists a secure w-dominating function on G if and only if w<sup>l</sup>* ≤ *lδ*.

**Proof.** If *f* is a secure *w*-dominating function on *G*, then *f* is a *w*-dominating function, and by Remark 1 we conclude that *w<sup>l</sup>* ≤ *lδ*.

Conversely, if *w<sup>l</sup>* ≤ *lδ*, then the function *f* , defined by *f*(*v*) = *l* for every *v* ∈ *V*(*G*), is a secure *w*-dominating function. Therefore, the result follows.

It was shown by Cabrera et al. [1] that the *w*-domination numbers satisfy a certain monotonicity. Given two integer vectors *w* = (*w*0, . . . , *w<sup>l</sup>* ) and *w* 0 = (*w* 0 0 , . . . , *w* 0 *l* ), we say that *w* <sup>0</sup> ≺ *w* if *w* 0 *<sup>i</sup>* ≤ *w<sup>i</sup>* for every *i* ∈ {0, . . . , *l*}. With this notation in mind, we can state the next remark which is a direct consequence of the definition of *w*-dominating function.

**Remark 3.** [1] *Let G be a graph of minimum degree δ and let w* = (*w*0, . . . , *w<sup>l</sup>* ), *w* 0 = (*w* 0 0 , . . . , *w* 0 *l* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w<sup>i</sup>* ≥ *wi*+<sup>1</sup> *and w* 0 *<sup>i</sup>* ≥ *w* 0 *i*+1 *for every i* ∈ {0, . . . , *l* − 1} *. If w* <sup>0</sup> ≺ *w and w<sup>l</sup>* ≤ *lδ, then every w-dominating function is a w*0 *-dominating function and, as a consequence,*

$$
\gamma\_{w'}(G) \le \gamma\_w(G).
$$

The monotonicity also holds for the case of secure *w*-domination.

**Remark 4.** *Let G be a graph of minimum degree δ and let w* = (*w*0, . . . , *w<sup>l</sup>* ), *w* 0 = (*w* 0 0 , . . . , *w* 0 *l* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w<sup>i</sup>* ≥ *wi*+<sup>1</sup> *and w* 0 *<sup>i</sup>* ≥ *w* 0 *i*+1 *for every i* ∈ {0, . . . , *l* − 1} *. If w* <sup>0</sup> ≺ *w and w<sup>l</sup>* ≤ *lδ, then every secure w-dominating function is a secure w*0 *-dominating function and, as a consequence,*

$$
\gamma\_{w'}^s(G) \le \gamma\_w^s(G).
$$

**Proof.** For any *γ s <sup>w</sup>*(*G*)-function *f* and any *v* ∈ *V*(*G*) with *f*(*v*) = 0, there exists *u* ∈ *M<sup>f</sup>* (*v*). Since *f* and *fu*→*<sup>v</sup>* are *w*-dominating functions, by Remark 3, we conclude that, if *w* <sup>0</sup> ≺ *w* and *w<sup>l</sup>* ≤ *lδ*, then both *f* and *fu*→*<sup>v</sup>* are *w* 0 -dominating functions. Therefore, *f* is a secure *w* 0 -dominating function and, as a consequence, *γ s <sup>w</sup>*0(*G*) ≤ *ω*(*f*) = *γ s <sup>w</sup>*(*G*).

From the following equality chain, we obtain examples of equalities in Remark 4. Graph *G*<sup>1</sup> is illustrated in Figure 1.

$$
\gamma^s\_{(3,0,0)}(G\_1) = \gamma^s\_{(3,1,0)}(G\_1) = \gamma^s\_{(3,2,0)}(G\_1) = \gamma^s\_{(3,2,1)}(G\_1) = \gamma^s\_{(3,2,2)}(G\_1).
$$

**Theorem 1.** *Let G be a graph of minimum degree δ, and let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w<sup>i</sup>* ≥ *wi*+<sup>1</sup> *for every i* ∈ {0, . . . , *l* − 1}*. If lδ* ≥ *w<sup>l</sup> , then the following statements hold.*

$$\text{(i)}\quad\gamma\_w(G)\le\gamma\_w^s(G).$$

(ii) *If k* ∈ Z <sup>+</sup>*, then <sup>γ</sup>*(*k*+1,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*) ≤ *γ s* (*k*,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*)*.*

**Proof.** Since every secure *w*-dominating function on *G* is a *w*-dominating function on *G*, (i) follows.

Let *f*(*V*0, . . . , *V<sup>l</sup>* ) be a *γ s* (*k*,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*)-function. Since *f* is a (*k*, *k* = *w*1, . . . , *w<sup>l</sup>* )-dominating function, *f*(*N*(*v*)) ≥ *w<sup>i</sup>* for every *v* ∈ *V<sup>i</sup>* with *i* ∈ {1, . . . , *l*} and *w*<sup>1</sup> = *k*. If *V*<sup>0</sup> = ∅, then *f* is a (*k* + 1, *k* = *w*1, . . . , *w<sup>l</sup>* )-dominating function, which implies that *γ*(*k*+1,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*) ≤ *ω*(*f*) = *γ s* (*k*,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*). Assume *V*<sup>0</sup> 6= ∅. Let *v* ∈ *V*<sup>0</sup> and *u* ∈ *M<sup>f</sup>* (*v*). If *f*(*N*(*v*)) = *k*, then *fu*→*v*(*N*(*v*)) = *f*(*N*(*v*)) − 1 = *k* − 1, which is a contradiction. Thus, *f*(*N*(*v*)) ≥ *k* + 1, which implies that *f* is a (*k* + 1, *k* = *w*1, . . . , *w<sup>l</sup>* )-dominating function. Therefore, *γ*(*k*+1,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*) ≤ *ω*(*f*) = *γ s* (*k*,*k*=*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*), and (ii) follows.

The inequalities above are tight. For instance, for any integers *n*, *n* <sup>0</sup> ≥ 4, we have that *γ*(2,2,2) (*K<sup>n</sup>* + *N<sup>n</sup>* <sup>0</sup>) = *γ s* (2,2,2) (*K<sup>n</sup>* + *N<sup>n</sup>* <sup>0</sup>) = 3 and *γ*(3,2,2) (*K*2,*n*) = *γ s* (2,2,2) (*K*2,*n*) = 5.

**Corollary 1.** *Let G be a graph of minimum degree δ and order n. Let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w<sup>i</sup>* ≥ *wi*+<sup>1</sup> *for every i* ∈ {0, . . . , *l* − 1} *and lδ* ≥ *w<sup>l</sup> . The following statements hold.*


**Proof.** Assume *n* > *w*0. By Theorem 1, we have that *γ s <sup>w</sup>*(*G*) ≥ *γw*(*G*). Now, if *γw*(*G*) ≤ *w*<sup>0</sup> −1 < *n* −1, then for any *γw*(*G*)-function *f* there exists at least one vertex *x* ∈ *V*(*G*) such that *f*(*x*) = 0 and *f*(*N*(*x*)) ≤ *ω*(*f*) < *w*0, which is a contradiction. Thus, *γ s <sup>w</sup>*(*G*) ≥ *γw*(*G*) ≥ *w*0.

Analogously, if *w*<sup>0</sup> = *w*1, then Theorem 1 leads to *γ s <sup>w</sup>*(*G*) ≥ *γ*(*w*0+1,*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*). In this case, if *γ*(*w*0+1,*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*) ≤ *w*<sup>0</sup> < *n*, then for any *γ*(*w*0+1,*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*)-function *f* there exists at least one vertex *x* ∈ *V*(*G*) such that *f*(*x*) = 0 and *f*(*N*(*x*)) ≤ *ω*(*f*) < *w*<sup>0</sup> + 1, which is a contradiction. Therefore, *γ s <sup>w</sup>*(*G*) ≥ *γ*(*w*0+1,*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*) ≥ *w*<sup>0</sup> + 1.

As the following result shows, the bounds above are tight.

**Proposition 1.** *For any integer n and any w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w<sup>l</sup>* ≤ · · · ≤ *w*<sup>0</sup> < *n,*

$$\gamma\_w^s(K\_\hbar) = \begin{cases} \ w\_0 + 1 & \text{if } w\_0 = w\_1, \\\ w\_0 & \text{otherwise.} \end{cases}$$

**Proof.** Assume *n* > *w*0. Let *S* ⊆ *V*(*Kn*) such that |*S*| = *w*<sup>0</sup> + 1 if *w*<sup>0</sup> = *w*<sup>1</sup> and |*S*| = *w*<sup>0</sup> otherwise. In both cases, the function *f*(*V*0, . . . , *V<sup>l</sup>* ), defined by *V*<sup>1</sup> = *S*, *V*<sup>0</sup> = *V*(*G*) \ *V*<sup>1</sup> and *V<sup>j</sup>* = ∅ whenever *j* 6∈ {0, 1}, is a secure *w*-dominating function. Hence, *γ s <sup>w</sup>*(*Kn*) ≤ *ω*(*f*) = |*S*|. Therefore, by Corollary 1 the result follows.

**Theorem 2.** *Let G be a graph of minimum degree δ, and let w* = (*w*0, . . . , *w<sup>l</sup>* ), *w* 0 = (*w* 0 0 , . . . , *w* 0 *l* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that lδ* ≥ *w<sup>l</sup> , w<sup>i</sup>* ≥ *wi*+<sup>1</sup> *and w* 0 *<sup>i</sup>* ≥ *w* 0 *i*+1 *for every i* ∈ {0, . . . , *l* − 1}*. If w<sup>i</sup>* ≥ *w* 0 *<sup>i</sup>*−<sup>1</sup> <sup>−</sup> <sup>1</sup> *for every i* ∈ {1, . . . , *l*}*, and* max{*w<sup>j</sup>* − 1, 0} ≥ *w* 0 *j for every j* ∈ {0, . . . , *l*}*, then*

$$
\gamma\_{w'}^s(G) \le \gamma\_w(G).
$$

**Proof.** Assume that *w<sup>i</sup>* ≥ *w* 0 *<sup>i</sup>*−<sup>1</sup> <sup>−</sup> <sup>1</sup> for every *<sup>i</sup>* ∈ {1, . . . , *<sup>l</sup>*} and max{*w<sup>j</sup>* <sup>−</sup> 1, 0} ≥ *<sup>w</sup>* 0 *j* for every *j* ∈ {0, . . . , *l*}. Let *f*(*V*0, . . . , *V<sup>l</sup>* ) be a *γw*(*G*)-function. We claim that *f* is a secure *w* 0 -dominating function. Since *f*(*N*(*x*)) ≥ *w<sup>i</sup>* ≥ *w* 0 *i* for every *x* ∈ *V<sup>i</sup>* with *i* ∈ {0, . . . , *l*}, we deduce that *f* is a *w* 0 -dominating function. Now, let *v* ∈ *V*<sup>0</sup> and *u* ∈ *N*(*v*) ∩ *V<sup>i</sup>* with *i* ∈ {1, . . . , *l*}. We differentiate the following cases for *x* ∈ *V*(*G*).

Case 1. *x* = *v*. In this case, *fu*→*v*(*v*) = 1 and *fu*→*v*(*N*(*v*)) = *f*(*N*(*v*)) − 1 ≥ *w*<sup>0</sup> − 1 ≥ max{*w*<sup>1</sup> − 1, 0} ≥ *w* 0 1 .

Case 2. *x* = *u*. In this case, *fu*→*v*(*u*) = *f*(*u*) − 1 = *i* − 1 and *fu*→*v*(*N*(*u*)) = *f*(*N*(*u*)) + 1 ≥ *w<sup>i</sup>* + 1 ≥ *w* 0 *i*−1 .

Case 3. *x* ∈ *V*(*G*) \ {*u*, *v*}. Assume *x* ∈ *V<sup>j</sup>* . Notice that *fu*→*v*(*x*) = *f*(*x*) = *j*. Now, if *x* 6∈ *N*(*u*) or *x* ∈ *N*(*u*) ∩ *N*(*v*), then *fu*→*v*(*N*(*x*)) = *f*(*N*(*x*)) ≥ *w<sup>j</sup>* ≥ *w* 0 *j* , while if *x* ∈ *N*(*u*) \ *N*[*v*], then *fu*→*v*(*N*(*x*)) = *f*(*N*(*x*)) − 1 ≥ max{*w<sup>j</sup>* − 1, 0} ≥ *w* 0 *j* .

According to the three cases above, *fu*→*<sup>v</sup>* is a *w* 0 -dominating function. Therefore, *f* is a secure *w* 0 -dominating function, and so *γ s <sup>w</sup>*0(*G*) ≤ *ω*(*f*) = *γw*(*G*).

The inequality above is tight. For instance, *γ s* (1,1,1) (*Kn*,*<sup>n</sup>* <sup>0</sup>) = *γ*(2,2,2) (*Kn*,*<sup>n</sup>* <sup>0</sup>) = 4 for *n*, *n* <sup>0</sup> ≥ 4.

From Theorems 1 and 2, we derive the next known inequality chain, where *G* has minimum degree *δ* ≥ 1, except in the last inequality in which *δ* ≥ 2.

$$
\gamma\_{\mathbf{s}}(\mathbf{G}) \le \gamma\_{\mathbf{2}}(\mathbf{G}) \le \gamma\_{\times 2}(\mathbf{G}) \le \gamma\_{\mathbf{st}}(\mathbf{G}) \le \gamma\_{\times 2, t}(\mathbf{G}).
$$

The following result is a particular case of Theorem 2.

**Corollary 2.** *Let G be a graph of minimum degree δ, and let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> and 1* = (1, . . . , 1)*. If* 0 ≤ *wj*−<sup>1</sup> − *w<sup>j</sup>* ≤ 2 *for every j* ∈ {1, . . . , *i*}*, where* 1 ≤ *i* ≤ *l and lδ* ≥ *w<sup>l</sup>* + 1*, then*

$$
\gamma\_{(w\_0,\ldots,w\_i,0,\ldots,0)}^s(G) \le \gamma\_{(w\_0+1,\ldots,w\_i+1,0,\ldots,0)}(G) \le \gamma\_{w+1}(G).
$$

For Graph *G*<sup>2</sup> illustrated in Figure 1, we have that *γ s* (1,1) (*G*2) = *γ s* (1,1,0) (*G*2) = *γ*(2,2,0) (*G*2) = *γ s* (1,1,1) (*G*2) = *γ*(2,2,2) (*G*2) = 3. Notice that *γ s <sup>w</sup>*(*G*2) = *γw*+**1**(*G*2) for *w* = **1** = (1, 1, 1).

Next, we show a class of graphs where *γw*(*G*) = *γw*+**1**(*G*). To this end, we need to introduce some additional notation and terminology. Given the two Graphs *G*<sup>1</sup> and *G*2, the *corona product graph G*<sup>1</sup> *G*<sup>2</sup> is the graph obtained from *G*<sup>1</sup> and *G*2, by taking one copy of *G*<sup>1</sup> and |*V*(*G*1)| copies of *G*<sup>2</sup> and joining by an edge every vertex from the *i*th copy of *G*<sup>2</sup> with the *i*th vertex of *G*1. For every *x* ∈ *V*(*G*1), the copy of *G*<sup>2</sup> in *G*<sup>1</sup> *G*<sup>2</sup> associated to *x* is denoted by *G*2,*x*.

**Theorem 3** ([1])**.** *Let G*<sup>1</sup> *G*<sup>2</sup> *be a corona graph where G*<sup>1</sup> *does not have isolated vertices, and let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> . If l* ≥ *w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> and* |*V*(*G*2)| ≥ *w*0*, then*

$$
\gamma\_w(\mathcal{G}\_1 \odot \mathcal{G}\_2) = w\_0 |V(\mathcal{G}\_1)|.
$$

From the result above, we deduce that under certain additional restrictions on *G*<sup>2</sup> and *w* we can obtain *γ s <sup>w</sup>*(*G*<sup>1</sup> *G*2) = *γw*+**1**(*G*<sup>1</sup> *G*2).

**Theorem 4.** *Let G*<sup>1</sup> *G*<sup>2</sup> *be a corona graph, where G*<sup>1</sup> *does not have isolated vertices and G*<sup>2</sup> *is a triangle-free graph. Let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that l* − 1 ≥ *w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> . If* |*V*(*G*2)| ≥ *w*<sup>0</sup> + 2*, then*

$$
\gamma\_w^s(\mathcal{G}\_1 \odot \mathcal{G}\_2) = (w\_0 + 1)|V(\mathcal{G}\_1)| = \gamma\_{w+1}(\mathcal{G}\_1 \odot \mathcal{G}\_2).
$$

**Proof.** Since *G*<sup>1</sup> does not have isolated vertices, the upper bound *γ s <sup>w</sup>*(*G*<sup>1</sup> *G*2) ≤ (*w*<sup>0</sup> + 1)|*V*(*G*1)| is straightforward, as the function *f* , defined by *f*(*x*) = *w*<sup>0</sup> + 1 for every *x* ∈ *V*(*G*1) and *f*(*x*) = 0 for the remaining vertices of *G*<sup>1</sup> *G*2, is a secure *w*-dominating function.

On the other hand, let *f*(*V*0, . . . , *V<sup>l</sup>* ) be a *γ s <sup>w</sup>*(*G*<sup>1</sup> *G*2)-function and suppose that there exists *x* ∈ *V*(*G*1) such that *f*(*V*(*G*2,*x*)) + *f*(*x*) ≤ *w*0. Since |*V*(*G*2,*x*)| ≥ *w*<sup>0</sup> + 2, there exist at least two different vertices *u*, *v* ∈ *V*(*G*2,*x*) ∩ *V*0. Hence, *f*(*N*(*u*)) = *f*(*N*(*v*)) = *w*0, which implies that *u* and *v* are adjacent and, since *G*<sup>2</sup> is a triangle-free graph, *f*(*x*) = *w*<sup>0</sup> and *f*(*y*) = 0 for every *y* ∈ *V*(*G*2,*x*). Thus, by Lemma 1, we conclude that *G*2,*<sup>x</sup>* is a clique, which is a contradiction as |*V*(*G*2)| ≥ 3 and *G*<sup>2</sup> is a triangle-free graph. This implies that *f*(*V*(*G*2,*x*)) + *f*(*x*) ≥ *w*<sup>0</sup> + 1 for every *x* ∈ *V*(*G*1), and so *γ s <sup>w</sup>*(*G*<sup>1</sup> *G*2) = *ω*(*f*) ≥ (*w*<sup>0</sup> + 1)|*V*(*G*1)|.

Therefore, *γ s <sup>w</sup>*(*G*<sup>1</sup> *G*2) = (*w*<sup>0</sup> + 1)|*V*(*G*1)|, and by Theorem 3 we conclude that *γw*+**1**(*G*<sup>1</sup> *G*2) = (*w*<sup>0</sup> + 1)|*V*(*G*1)|, which completes the proof.

**Theorem 5.** *Let <sup>G</sup> be a graph of minimum degree <sup>δ</sup> and <sup>l</sup>* ≥ <sup>2</sup> *an integer. For any* (*w*0, . . . , *<sup>w</sup>l*−<sup>1</sup> ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*l*−<sup>1</sup> *with w*<sup>0</sup> ≥ · · · ≥ *wl*−<sup>1</sup> *and lδ* ≥ *wl*−<sup>1</sup> *,*

$$
\gamma\_{(w\_0, \ldots, w\_{l-1}, w\_l = w\_{l-1})}^s(G) \le \gamma\_{(w\_0, \ldots, w\_{l-1})}(G) + \gamma(G).
$$

**Proof.** Let *<sup>f</sup>*(*V*0, . . . , *<sup>V</sup>l*−<sup>1</sup> ) be a *<sup>γ</sup>*(*w*0,...,*wl*−<sup>1</sup> ) (*G*)-function and *S* a *γ*(*G*)-set. We define a function *g*(*W*0, . . . , *W<sup>l</sup>* ) as follows. Let *<sup>W</sup><sup>l</sup>* = *<sup>V</sup>l*−<sup>1</sup> ∩ *<sup>S</sup>*, *<sup>W</sup>*<sup>0</sup> = *<sup>V</sup>*<sup>0</sup> \ *<sup>S</sup>*, and *<sup>W</sup><sup>i</sup>* = (*Vi*−<sup>1</sup> ∩ *<sup>S</sup>*) ∪ (*V<sup>i</sup>* \ *<sup>S</sup>*) for every *i* ∈ {1, . . . , *l* − 1}.

We claim that *<sup>g</sup>* is a secure (*w*0, . . . , *<sup>w</sup>l*−<sup>1</sup> , *<sup>w</sup><sup>l</sup>* = *<sup>w</sup>l*−<sup>1</sup> )-dominating function. First, we observe that, if *x* ∈ *W<sup>i</sup>* ∩ *S* with *i* ∈ {1, . . . , *l*}, then *x* ∈ *Vi*−<sup>1</sup> and *g*(*N*(*x*)) ≥ *f*(*N*(*x*)) ≥ *wi*−<sup>1</sup> ≥ *w<sup>i</sup>* . Moreover, if *x* ∈ *W<sup>i</sup>* \ *S* with *i* ∈ {0, . . . , *l* − 1}, then *x* ∈ *V<sup>i</sup>* and *g*(*N*(*x*)) ≥ *f*(*N*(*x*)) ≥ *w<sup>i</sup>* . Hence, *g* is a (*w*0, . . . , *<sup>w</sup>l*−<sup>1</sup> , *<sup>w</sup><sup>l</sup>* = *<sup>w</sup>l*−<sup>1</sup> )-dominating function.

Now, let *v* ∈ *W*<sup>0</sup> = *V*<sup>0</sup> \ *S*. Notice that there exists a vertex *u* ∈ *N*(*v*) ∩ *Vi*−<sup>1</sup> ∩ *S* with *i* ∈ {1, . . . , *l*}. Hence, *u* ∈ *N*(*v*) ∩ *W<sup>i</sup>* . We differentiate the following cases for *x* ∈ *V*(*G*).

Case 1. *x* = *v*. In this case, *gu*→*v*(*v*) = 1 and, as *N*(*v*) ∩ *S* 6= ∅, we obtain that *gu*→*v*(*N*(*v*)) = *g*(*N*(*v*)) − 1 ≥ *f*(*N*(*v*)) ≥ *w*<sup>0</sup> ≥ *w*1.

Case 2. *x* = *u*. In this case, *gu*→*v*(*u*) = *g*(*u*) − 1 = *i* − 1 and *gu*→*v*(*N*(*u*)) = *g*(*N*(*u*)) + 1 ≥ *f*(*N*(*u*)) + 1 ≥ *wi*−<sup>1</sup> + 1 > *wi*−<sup>1</sup> .

Case 3. *x* ∈ *V*(*G*) \ {*u*, *v*}. Assume *x* ∈ *W<sup>j</sup>* . Notice that *gu*→*v*(*x*) = *g*(*x*) = *j*. If *x* 6∈ *N*(*u*) or *x* ∈ *N*(*u*) ∩ *N*(*v*), then *gu*→*v*(*N*(*x*)) = *g*(*N*(*x*)) ≥ *f*(*N*(*x*)) ≥ *w<sup>j</sup>* .

Moreover, if *x* ∈ (*N*(*u*) \ *N*[*v*]) ∩ *S*, then *x* ∈ *Vj*−<sup>1</sup> and so *gu*→*v*(*N*(*x*)) = *g*(*N*(*x*)) − 1 ≥ *f*(*N*(*x*)) ≥ *wj*−<sup>1</sup> ≥ *w<sup>j</sup>* . Finally, if *x* ∈ (*N*(*u*) \ *N*[*v*]) \ *S*, then *x* ∈ *V<sup>j</sup>* and therefore *gu*→*v*(*N*(*x*)) = *g*(*N*(*x*)) − 1 ≥ *f*(*N*(*x*)) ≥ *w<sup>j</sup>* .

According to the three cases above, *<sup>g</sup>u*→*<sup>v</sup>* is a (*w*0, . . . , *<sup>w</sup>l*−<sup>1</sup> , *<sup>w</sup><sup>l</sup>* = *<sup>w</sup>l*−<sup>1</sup> )-dominating function. Therefore, *<sup>f</sup>* is a secure (*w*0, . . . , *<sup>w</sup>l*−<sup>1</sup> , *<sup>w</sup><sup>l</sup>* = *<sup>w</sup>l*−<sup>1</sup> )-dominating function, and so *γ s* (*w*0,...,*wl*−<sup>1</sup> ,*wl*=*wl*−<sup>1</sup> ) (*G*) <sup>≤</sup> *<sup>ω</sup>*(*g*) <sup>≤</sup> *<sup>ω</sup>*(*f*) + <sup>|</sup>*S*<sup>|</sup> <sup>=</sup> *<sup>γ</sup>*(*w*0,...,*wl*−<sup>1</sup> ) (*G*) + *γ*(*G*).

From Theorem 5, we derive the next known inequalities, which are tight.

**Corollary 3.** *For a graph G, the following statements hold.*


To establish the following result, we need to define the following parameter.

$$\nu^s\_{(\mathfrak{w}\_0,\dots,\mathfrak{w}\_l)}(G) = \max\{|V\_0| \colon f(V\_0,\dots,V\_l) \text{ is a } \gamma^s\_{(\mathfrak{w}\_0,\dots,\mathfrak{w}\_l)}(G)\text{-function.}\},$$

In particular, for *l* = 1 and a graph *G* of order *n*, we have that *ν s* (*w*0,*w*<sup>1</sup> ) (*G*) = *n* − *γ s* (*w*0,*w*<sup>1</sup> ) (*G*).

**Theorem 6.** *Let G be a graph of minimum degree δ and order n. The following statements hold for any* (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> with w*<sup>0</sup> ≥ · · · ≥ *<sup>w</sup><sup>l</sup> .*


**Proof.** If there exists *i* ∈ {1, . . . , *l* − 1} such that *iδ* ≥ *w<sup>i</sup>* , then for any *γ s* (*w*0,...,*w<sup>i</sup>* ) (*G*)-function *f*(*V*0, . . . , *Vi*) we define a secure (*w*0, . . . , *w<sup>l</sup>* )-dominating function *g*(*W*0, . . . , *W<sup>l</sup>* ) by *W<sup>j</sup>* = *V<sup>j</sup>* for every *j* ∈ {0, . . . , *i*} and *W<sup>j</sup>* = ∅ for every *j* ∈ {*i* + 1, . . . , *l*}. Hence, *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) ≤ *ω*(*g*) = *ω*(*f*) = *γ s* (*w*0,...,*w<sup>i</sup>* ) (*G*). Therefore, (i) follows.

Now, assume *l* ≥ *i* + 1 > *w*0. Let *S* be a *γ*(*G*)-set. Let *f* be the function defined by *f*(*v*) = *i* + 1 for every *v* ∈ *S* and *f*(*v*) = 0 for the remaining vertices. Since *i* + 1 > *w*0, we can conclude that *f* is a secure (*w*0, . . . , *w<sup>i</sup>* , 0 . . . , 0)-dominating function. Therefore, *γ s* (*w*0,...,*w<sup>i</sup>* ,0...,0) (*G*) ≤ *ω*(*f*) = (*i* + 1)|*S*| = (*i* + 1)*γ*(*G*), which implies that (ii) follows.

To prove (iii), assume that *l* ≥ *ki*, *iδ* ≥ *w* 0 *i* and *wkj* = *kw*<sup>0</sup> *j* for every *j* ∈ {0, . . . , *i*}. Let *f* 0 (*V* 0 0 , . . . , *V* 0 *i* ) be a *γ s* (*w*0 0 ,...,*w*0 *i* ) (*G*)-function. We construct a function *f*(*V*0, . . . , *V<sup>l</sup>* ) as *f*(*v*) = *k f* 0 (*v*) for every *v* ∈ *V*(*G*). Hence, *Vkj* = *V* 0 *j* for every *j* ∈ {0, . . . , *i*}, while *V<sup>j</sup>* = ∅ for the remaining cases. Thus, for every *v* ∈ *Vkj* with *j* ∈ {0, . . . , *i*} we have that *f*(*N*(*v*)) = *k f* <sup>0</sup> (*N*(*v*)) ≥ *kw*<sup>0</sup> *<sup>j</sup>* = *wkj*, which implies that *f* is a (*w*0, . . . , *w<sup>l</sup>* )-dominating function. Now, for every *x* ∈ *V*0, there exists *y* ∈ *M<sup>f</sup>* <sup>0</sup>(*x*). Hence, for every *v* ∈ *Vkj* with *j* ∈ {0, . . . , *i*}, we have that *fy*→*x*(*N*(*v*)) = *k f* <sup>0</sup> *y*→*x* (*N*(*v*)) ≥ *kw*<sup>0</sup> *<sup>j</sup>* = *wkj*, which implies that *fy*→*<sup>x</sup>* is a (*w*0, . . . , *w<sup>l</sup>* )-dominating function. Therefore, *f* is a secure (*w*0, . . . , *w<sup>l</sup>* )-dominating function, and so *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) ≤ *ω*(*f*) = *kω*(*f* 0 ) = *kγ s* (*w*0 0 ,...,*w*0 *i* ) (*G*). Therefore, (iii) follows.

Now, assume that *lδ* ≥ *k* + *w<sup>l</sup>* > *k* and *w*<sup>0</sup> + *k* ≥ *β*<sup>1</sup> ≥ · · · ≥ *β<sup>k</sup>* ≥ *w*<sup>1</sup> + *k*. Let *g*(*W*0, . . . , *W<sup>l</sup>* ) be a *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*)-function. We construct a function *f*(*V*0, . . . , *Vl*+*<sup>k</sup>* ) as *f*(*v*) = *g*(*v*) + *k* for every *v* ∈ *V*(*G*) \ *W*<sup>0</sup> and *f*(*v*) = 0 for every *v* ∈ *W*0. Hence, *Vj*+*<sup>k</sup>* = *W<sup>j</sup>* for every *j* ∈ {1, . . . , *l*}, *V*<sup>0</sup> = *W*<sup>0</sup> and *V<sup>j</sup>* = ∅ for the remaining cases. Thus, if *v* ∈ *Vj*+*<sup>k</sup>* and *j* ∈ {1, . . . , *l*}, then *f*(*N*(*v*)) ≥ *g*(*N*(*v*)) + *k* ≥ *w<sup>j</sup>* + *k*, and if *v* ∈ *V*0, then *f*(*N*(*v*)) ≥ *g*(*N*(*v*)) + *k* ≥ *w*<sup>0</sup> + *k*. This implies that *f* is a (*w*<sup>0</sup> + *k*, *β*1, . . . , *β<sup>k</sup>* , *w*<sup>1</sup> + *k*, . . . , *w<sup>l</sup>* + *k*)-dominating function. Now, for every *x* ∈ *V*<sup>0</sup> = *W*0, there exists *y* ∈ *Mg*(*x*). Hence, if *v* ∈ *Vj*+*<sup>k</sup>* and *j* ∈ {1, . . . , *l*}, then *fy*→*x*(*N*(*v*)) ≥ *gy*→*x*(*N*(*v*)) + *k* ≥ *w<sup>j</sup>* + *k*, and if *v* ∈ *V*0, then *fy*→*x*(*N*(*v*)) ≥ *gy*→*x*(*N*(*v*)) + *k* ≥ *w*<sup>0</sup> + *k*. This implies that *fy*→*<sup>x</sup>* is a (*w*<sup>0</sup> + *k*, *β*1, . . . , *β<sup>k</sup>* , *w*<sup>1</sup> + *k*, . . . , *w<sup>l</sup>* + *k*)-dominating function, and so *f* is a secure (*w*<sup>0</sup> + *k*, *β*1, . . . , *β<sup>k</sup>* , *w*<sup>1</sup> + *k*, . . . , *w<sup>l</sup>* + *k*)-dominating function. Therefore, *γ s* (*w*0+*k*,*β*<sup>1</sup> ,...,*β<sup>k</sup>* ,*w*1+*k*,...,*wl*+*k*) (*G*) ≤ *ω*(*f*) = *ω*(*g*) + *k* ∑ *l j*=1 |*W<sup>j</sup>* | = *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) + *k*(*n* − |*W*0|) ≤ *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) + *k*(*n* − *ν s* (*w*0,...,*w<sup>l</sup>* ) (*G*)), concluding that (iv) follows.

Furthermore, if *lδ* ≥ *w<sup>l</sup>* ≥ *l* ≥ 2, then, by applying (iv) for *k* = *l* − 1, we deduce that

$$\gamma\_{\left(\mathfrak{w}\_{0},\ldots,\mathfrak{w}\_{l}\right)}^{s}(\mathbf{G}) \leq \gamma\_{\left(\mathfrak{w}\_{0}-l+1,\mathfrak{w}\_{l}-l+1\right)}^{s}(\mathbf{G}) + (l-1)(n-\upsilon\_{\left(\mathfrak{w}\_{0}-l+1,\mathfrak{w}\_{l}-l+1\right)}^{s}(\mathbf{G})) = l\gamma\_{\left(\mathfrak{w}\_{0}-l+1,\mathfrak{w}\_{l}-l+1\right)}^{s}(\mathbf{G}).$$

Therefore, (v) follows.

In the next subsections, we consider several applications of Theorem 6 where we show that the bounds are tight. For instance, the following particular cases is of interest.

**Corollary 4.** *Let G be a graph of minimum degree δ, and let k*, *l*, *w*2, . . . , *w<sup>l</sup>* ∈ Z <sup>+</sup> *with k* <sup>≥</sup> *<sup>w</sup>*<sup>2</sup> ≥ · · · ≥ *<sup>w</sup><sup>l</sup> .*

$$\text{(i')}\quad \text{If } \delta \ge k \text{, then } \gamma\_{(k+1,k,w\_2,\dots,w\_l)}^s(G) \le \gamma\_{(k+1,k)}^s(G).$$

$$\text{(ii')}\quad \text{If } \delta \ge k \text{, then } \gamma^s\_{(k,k,w\_2,\ldots,w\_l)}(G) \le \gamma^s\_{(k,k)}(G).$$

$$\text{(iii)}\quad If \, l \delta \ge k \ge l \ge 2 \text{, then } \gamma^s\_{\underbrace{(k,k,\dots,k)}\_{l+1}}(\mathcal{G}) \le l \gamma^s\_{(k-l+1,k-l+1)}(\mathcal{G}).$$

$$\text{(iv')}\quad \text{Let } i \in \mathbb{Z}^+.\text{ If } l \ge \text{ki and } \delta \ge 1 \text{, then } \gamma^s\_{\underbrace{(k,\dots,k)}\_{l+1}}(\mathcal{G}) \le k \gamma^s\_{\underbrace{(1,\dots,1)}\_{i+1}}(\mathcal{G}).\text{}$$

**Proof.** If *δ* ≥ *k*, then by Theorem 6 (i) we conclude that (i') and (ii') follow. If *lδ* ≥ *k* ≥ *l* ≥ 2, then by Theorem 6 (v) we deduce (iii'). Finally, if *l* ≥ *k* and *δ* ≥ 1, then by Theorem 6 (iii) we deduce that (iv') follows.

To show that the inequalities above are tight, we consider the following examples. For (i'), we have *γ s* (2,1,1) (*K*<sup>1</sup> + (*K*<sup>2</sup> ∪ *K*2)) = *γ s* (2,1) (*K*<sup>1</sup> + (*K*<sup>2</sup> ∪ *K*2)) = 3. For (ii') we have *γ s* (*k*,*k*,*w*2,...,*w<sup>l</sup>* ) (*G*) = *γ s* (*k*,*k*) (*G*) = *k* + 1 for every graph *G* with *k* + 1 universal vertices. Finally, for (iii') and (iv'), we take *l* = *k* = 2 and *γ s* (2,2,2) (*K*<sup>2</sup> + *Nn*) = 2*γ s* (1,1) (*K*<sup>2</sup> + *Nn*) = 4 for every *n* ≥ 2.

We already know that *γt*(*G*) = *γ*(1,1) (*G*) = *γ*(1,1,*w*2,...,*w<sup>l</sup>* ) (*G*), for every *w*2, . . . , *w<sup>l</sup>* ∈ {0, 1}. In contrast, the picture is quite different for the case of secure (1, 1)-domination, as there are graphs

where the gap *γ s* (1,1) (*G*) − *γ s* (1,...,1) (*G*) is arbitrarily large. For instance, lim*n*→<sup>∞</sup> *γ s* (1,1) (*K*1,*n*−1) = +∞, while, if *<sup>l</sup>* <sup>≥</sup> 2, then lim *<sup>n</sup>*→+<sup>∞</sup> *γ s* (1, . . . , 1 | {z } *l*+1 ) (*K*1,*n*−1) = 3.

**Proposition 2.** *Let G be a graph of order n. Let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> . If G* 0 *is a spanning subgraph of G with minimum degree δ* <sup>0</sup> ≥ *wl l , then*

$$
\gamma\_w^s(G) \le \gamma\_w^s(G').
$$

**Proof.** Let *E* <sup>−</sup> = {*e*1, . . . ,*ek*} be the set of all edges of *G* not belonging to the edge set of *G* 0 . Let *G* 0 <sup>0</sup> = *G* and, for every *i* ∈ {1, . . . , *k*}, let *X<sup>i</sup>* = {*e*1, . . . ,*ei*} and *G* 0 *<sup>i</sup>* = *G* − *X<sup>i</sup>* , the edge-deletion subgraph of *G* induced by *E*(*G*) \ *X<sup>i</sup>* .

For any *γ s <sup>w</sup>*(*G* 0 *i* )-function *f* and any *v* ∈ *V*(*G* 0 *i* ) = *V*(*G*) with *f*(*v*) = 0, there exists *u* ∈ *M<sup>f</sup>* (*v*). Since *f* and *fu*→*<sup>v</sup>* are *w*-dominating functions on *G* 0 *i* , both are *w*-dominating functions on *G* 0 *i*−1 , and so we can conclude that *f* is a secure *w*-dominating function on *G* 0 *i*−1 , which implies that *γ s <sup>w</sup>*(*G* 0 *i*−1 ) ≤ *γ s <sup>w</sup>*(*G* 0 *i* ). Hence, *γ s <sup>w</sup>*(*G*) = *γ s <sup>w</sup>*(*G* 0 0 ) ≤ *γ s <sup>w</sup>*(*G* 0 1 ) ≤ · · · ≤ *γ s <sup>w</sup>*(*G* 0 *k* ) = *γ s <sup>w</sup>*(*G* 0 ).

As a simple example of equality in Proposition 2 we can take any graph *G* of order *n*, having *n* <sup>0</sup> + 1 ≥ 2 universal vertices. In such a case, for *n* <sup>0</sup> = *w*<sup>1</sup> ≥ · · · ≥ *w<sup>l</sup>* we have that

$$\gamma^{\mathcal{S}}\_{\left(\mathfrak{n}',\mathfrak{n}'=\mathfrak{w}\_1,\ldots,\mathfrak{w}\_l\right)}(\mathcal{K}\_{\mathfrak{n}}) = \gamma^{\mathcal{S}}\_{\left(\mathfrak{n}',\mathfrak{n}'=\mathfrak{w}\_1,\ldots,\mathfrak{w}\_l\right)}(\mathcal{G}) = \gamma^{\mathcal{S}}\_{\left(\mathfrak{n}',\mathfrak{n}'\right)}(\mathcal{K}\_{\mathfrak{n}}) = \gamma^{\mathcal{S}}\_{\left(\mathfrak{n}',\mathfrak{n}'\right)}(\mathcal{G}) = \mathfrak{n}'+1.$$

From Proposition 2, we obtain the following result.

**Corollary 5.** *Let G be a graph of order n and w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> .*


To derive some lower bounds on *γ s <sup>w</sup>*(*G*), we need to establish the following lemma.

**Lemma 2** ([1])**.** *Let G be a graph with no isolated vertex, maximum degree* ∆ *and order n. For any w-dominating function f*(*V*0, . . . , *V<sup>l</sup>* ) *on G such that w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> ,*

$$
\Delta\omega(f) \ge w\_0 n + \sum\_{i=1}^{l} (w\_i - w\_0)|V\_i|.
$$

**Theorem 7.** *Let G be a graph with no isolated vertex, maximum degree* ∆ *and order n. Let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> and lδ* ≥ *w<sup>l</sup> . The following statements hold.*

• *If w*<sup>0</sup> = *w*<sup>1</sup> *and w*<sup>0</sup> − *w<sup>i</sup>* ≤ *i for every i* ∈ {2, ..., *l*}*, then γ s <sup>w</sup>*(*G*) ≥ l (*w*0+1)*n* ∆+1 m .

$$\bullet \quad \text{If } w\_0 = w\_1 \text{, then } \gamma\_w^s(G) \ge \left\lceil \frac{(w\_0+1)n}{\Delta + w\_0} \right\rceil$$

• *If w*<sup>0</sup> = *w*<sup>1</sup> + 1 *and w*<sup>0</sup> − *w<sup>i</sup>* ≤ *i for every i* ∈ {2, ..., *l*}*, then γ s <sup>w</sup>*(*G*) ≥ *w*0*n* ∆+1 .

.

• *γ s <sup>w</sup>*(*G*) ≥ l *w*0*n* ∆+*w*<sup>0</sup> m . **Proof.** Let *w*<sup>0</sup> = *w*<sup>1</sup> and *w*<sup>0</sup> − *w<sup>i</sup>* ≤ *i* for every *i* ∈ {2, ..., *l*}. Let *f*(*V*0, . . . , *V<sup>l</sup>* ) be a *γ*(*w*0+1,*w*<sup>1</sup> ,...,*w<sup>l</sup>* ) (*G*)-function. By Lemma 2, we deduce the following.

$$\begin{aligned} \Delta\omega(f) &\geq (w\_0+1)n + \sum\_{i=1}^l (w\_i - w\_0)|V\_i| \\ &\geq (w\_0+1)n - \sum\_{i=1}^l i|V\_i| \\ &= (w\_0+1)n - \omega(f). \end{aligned}$$

Therefore, Theorem 1 (ii) leads to *γ s <sup>w</sup>*(*G*) ≥ *ω*(*f*) ≥ l (*w*0+1)*n* ∆+1 m .

The proof of the remaining items is completely analogous. In the last two cases, we consider that *f*(*V*0, . . . , *V<sup>l</sup>* ) is a *γw*(*G*)-function, and we apply Theorem 1 (i) instead of (ii).

The bounds above are sharp. For instance, *γ s* (1,1,0) (*G*) ≥ 2*n* ∆+1 is achieved by Graph *G*<sup>2</sup> shown in Figure 1, the bound *γ s* (*k*,*k*,0) (*G*) ≥ l (*k*+1)*n* ∆+*k* m is achieved by *G* ∼= *K<sup>n</sup>* for every *n* > *k*(*k* −1) > 0, the bound *γ s* (2,1,1) (*G*) ≥ 2*n* ∆+1 is achieved by the corona graph *K*<sup>2</sup> *K<sup>n</sup>* <sup>0</sup> with *n* <sup>0</sup> ≥ 4, while *γ s* (2,0,0) (*G*) ≥ 2*n* ∆+2 is achieved by *G* ∼= *C*5, *G* ∼= *K<sup>n</sup>* and *G* ∼= *K<sup>n</sup>* <sup>0</sup> ∪ *K<sup>n</sup>* <sup>0</sup> with *n* ≥ 2 and *n* <sup>0</sup> ≥ 4.

To conclude the paper, we consider the problem of characterizing the graphs *G* and the vectors *w* for which *γ s <sup>w</sup>*(*G*) takes small values. It is readily seen that *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) = 1 if and only if *w*<sup>0</sup> = 1, *w*<sup>1</sup> = 0 and *G* ∼= *Kn*. Next, we consider the case *γ s <sup>w</sup>*(*G*) = 2.

**Theorem 8.** *Let w* = (*w*0, . . . , *w<sup>l</sup>* ) ∈ Z <sup>+</sup> <sup>×</sup> <sup>N</sup>*<sup>l</sup> such that w*<sup>0</sup> ≥ · · · ≥ *w<sup>l</sup> . For a graph G of order at least three, γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) = 2 *if and only if one of the following conditions holds.*

(i) *w*<sup>2</sup> = 0*, γ*(*G*) = 1 *and one of the following conditions holds.*

(1,0)

(1,1)

$$\bullet \qquad w\_0 = w\_1 = 1.$$


**Proof.** Assume first that *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) = 2 and let *f*(*V*0, . . . , *V<sup>l</sup>* ) be a *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*)-function. Notice that (*w*0, *w*1) ∈ {(1, 0),(1, 1),(2, 0),(2, 1)} and |*V*2| ∈ {0, 1}.

Firstly, we consider that |*V*2| = 1, i.e., *V*<sup>2</sup> = {*u*} for some universal vertex *u* ∈ *V*(*G*). In this case, *w*<sup>2</sup> = 0, *γ*(*G*) = 1, and *V<sup>i</sup>* = ∅ for every *i* 6= 0, 2. By Lemma 1, if *w*<sup>0</sup> = 2, then *G*[*T<sup>f</sup>* (*u*)] = *G*[*V*(*G*) \ {*u*}] is a clique, which implies that *G* ∼= *Kn*. Obviously, in such a case, *w*<sup>1</sup> < 2. Finally, the case, *w*<sup>0</sup> = 1 and *w*<sup>1</sup> = 0 leads to *G* 6∼= *Kn*, as *γ s* (1,0...,0) (*Kn*) = 1. Therefore, (i) follows.

From now on, assume that *V*<sup>2</sup> = ∅. Hence, *V<sup>i</sup>* = ∅ for every *i* 6= 0, 1. If *w*<sup>0</sup> = 1 and *w*<sup>1</sup> = 0, then *G* 6∼= *K<sup>n</sup>* and *V*<sup>1</sup> is a secure dominating set. Therefore, (ii) follows. If *w*<sup>0</sup> = *w*<sup>1</sup> = 1, then *V*<sup>1</sup> is a secure total dominating set of cardinality two, and so *γ s* (1,1) (*G*) = 2. Therefore, (iii) follows. Finally, assume *w*<sup>0</sup> = 2. In this case, *V*<sup>1</sup> is a double dominating set of cardinality two, and by Lemma 1 we know that *G*[*T<sup>f</sup>* (*x*)] = *G*[*V*(*G*) \ *V*1] is a clique for any *x* ∈ *V*1. Hence, *G* ∼= *K<sup>n</sup>* and, in such a case, *w*<sup>1</sup> < 2. Therefore, (iv) follows.

Conversely, if one of the four conditions holds, then it is easy to check that *γ s* (*w*0,...,*w<sup>l</sup>* ) (*G*) = 2, which completes the proof.

**Author Contributions:** All authors contributed equally to this work. Investigation, A.C.M., A.E.-M., and J.A.R.-V.; and Writing—review and editing, A.C.M., A.E.-M., and J.A.R.-V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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## *Article* **Quasi-Ordinarization Transform of a Numerical Semigroup**

**Maria Bras-Amorós \* , Hebert Pérez-Rosés and José Miguel Serradilla-Merinero**

Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain; hebert.perez@urv.cat (H.P.-R.); josemiguel.serradilla@estudiants.urv.cat (J.M.S.-M.) **\*** Correspondence: maria.bras@urv.cat

**Abstract:** In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms.

**Keywords:** numerical semigroup; forest; ordinarization transform; quasi-ordinarization transform

## **1. Introduction**

A numerical semigroup is a cofinite submonoid of N<sup>0</sup> under addition, where N<sup>0</sup> is the set of nonnegative integers.

While the symmetry of structures has traditionally been studied with the aid of groups, it is also possible to relax the definition of symmetry, so as to describe some forms of symmetry that arise in quasicrystals, fractals, and other natural phenomena, with the aid of semigroups or monoids, rather than groups. For example, Rosenfeld and Nordahl [1] lay the groundwork for such a theory of symmetry based on semigroups and monoids, and they cite some applications in chemistry.

Suppose that Λ is a numerical semigroup. The elements in the complement N<sup>0</sup> \ Λ are called the *gaps* of the semigroup and the number of gaps is its *genus*. The *Frobenius number* is the largest gap and the *conductor* is the non-gap that equals the Frobenius number plus one. The first non-zero non-gap of a numerical semigroup (usually denoted by *m*) is called its *multiplicity*. An *ordinary* semigroup is a numerical semigroup different from N<sup>0</sup> in which all gaps are in a row. The non-zero non-gaps of a numerical semigroup that are not the result of the sum of two smaller non-gaps are called the *generators* of the numerical semigroup. It is easy to deduce that the set of generators of a numerical semigroup must be co-prime. One general reference for numerical semigroups is [2].

To illustrate all these definitions, consider the well-tempered harmonic semigroup *H* = {0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . }, where we use " . . . " to indicate that the semigroup consecutively contains all the integers from the number that precedes the ellipsis. The semigroup *H* arises in the mathematical theory of music [3]. It is obviously cofinite and it contains zero. One can also check that it is closed under addition. Hence, it is a numerical semigroup. Its Frobenius number is 44, its conductor is 45, its genus is 33, and its multiplicity is 12. Its generators are {12, 19, 28, 34, 42, 45, 49, 51}.

The number of numerical semigroups of genus *g* is denoted *ng*. It was conjectured in [4] that the sequence *n<sup>g</sup>* asymptotically behaves as the Fibonacci numbers. In particular, it was conjectured that each term in the sequence is larger than the sum of the two previous terms, that is, *n<sup>g</sup>* > *ng*−<sup>1</sup> + *ng*−<sup>2</sup> for *g* > 2, with each term being increasingly similar to the sum of the two previous terms as *g* approaches infinity, more precisely lim*g*→<sup>∞</sup> *ng <sup>n</sup>g*−1+*ng*−<sup>2</sup> = 1 and, equivalently, lim*g*→<sup>∞</sup> *ng ng*−<sup>1</sup> = *φ* = 1+ √ 5 2 . A number of papers deal with the sequence

**Citation:** Bras-Amorós, M.; Pérez-Rosés, H.; Serradilla-Merinero, J.M. Quasi-Ordinarization Transform of a Numerical Semigroup. *Symmetry* **2021**, *13*, 1084. https://doi.org/ 10.3390/sym13061084

Academic Editor: Michel Planat

Received: 9 February 2021 Accepted: 4 June 2021 Published: 17 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

semigroup Λ0

tree up to depth 7.

a semigroup is the genus of the semigroup.

*n<sup>g</sup>* [5–20]. Alex Zhai proved the asymptotic Fibonacci-like behavior of *n<sup>g</sup>* [21]. However, it remains unproven that *n<sup>g</sup>* is increasing. This was already conjectured by Bras-Amorós in [22]. More information on *ng*, as well as the list of the first 73 terms can be found in entry A007323 of The On-Line Encyclopedia of Integer Sequences [23].

It is well known that all numerical semigroups can be organized in an infinite tree T whose root is the semigroup N<sup>0</sup> and in which the parent of a numerical semigroup Λ is the numerical semigroup Λ0 obtained by adjoining to Λ its Frobenius number. For instance, the parent of the semigroup *H* = {0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . } is the semigroup *H*<sup>0</sup> = {0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, **44**, 45, 46, 47, 48, . . . }. In turn, the children of a numerical semigroup are the semigroups we obtain by taking away the generators one by one that are larger than or equal to the conductor of the semigroup. The parent of a numerical semigroup of genus *g* has genus *g* − 1 and all numerical semigroups are in T, at a depth equal to its genus. In particular, *n<sup>g</sup>* is the number of nodes of T at depth *g*. This construction was already considered in [24]. Figure 1 shows the tree up to depth 7. Version June 3, 2021 submitted to *Journal Not Specified* 2 of 17

**Figure 1.** The tree T up to depth 7. White dots refer to the gaps, dark gray dots to the generators and the light gray ones to the elements of the semigroups that are not generators. **Figure 1.** The tree T up to depth 7. White dots refer to the gaps, dark gray dots to the generators and the light gray ones to the elements of the semigroups that are not generators.

 The number of numerical semigroups of genus *g* is denoted *ng*. It was conjectured in [5] that the sequence *n<sup>g</sup>* asymptotically behaves as the Fibonacci numbers. In particular, it was conjectured that each term in the sequence is larger than the sum of the two previous terms, that is, *n<sup>g</sup>* > *ng*−<sup>1</sup> + *ng*−<sup>2</sup> for *g* > 2, being each term more and more similar to the sum of the two previous terms as *g* approaches infinity, more precisely lim*g*→<sup>∞</sup> *ng* = 1 and, equivalently, lim*g*→<sup>∞</sup> *ng* = *φ* = + √ . A number In [9], a new tree construction is introduced as follows. The *ordinarization transform* of a non-ordinary semigroup Λ with Frobenius number *F* and multiplicity *m* is the set Λ<sup>0</sup> = Λ \ {*m*} ∪ {*F*}. For instance, the ordinarization transform of the semigroup *H* = {0, **12**, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . } is the semigroup *H*<sup>0</sup> = {0, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, **44**, 45, 46, 47, 48, . . . } The ordinarization transform of an ordinary semigroup is then defined to be itself. Note that the genus of the ordinarization transform of a semigroup is the genus of the semigroup.

*<sup>n</sup>g*−1+*ng*−<sup>2</sup> *ng*−<sup>1</sup> of papers deal with the sequence *n<sup>g</sup>* [1–3,6,7,9–11,13,16–20,22,28]. Alex Zhai proved the asymptotic Fibonacci-like behavior of *n<sup>g</sup>* [27]. However, it remains not proved that *n<sup>g</sup>* is increasing. This was already conjectured by Bras-Amorós in [4]. More information on *ng*, as well as the list of the first 73 terms can be found in entry A007323 of The On-Line Encyclopedia of Integer Sequences [23]. It is well known that all numerical semigroups can be organized in an infinite tree T whose root is the semigroup N<sup>0</sup> and in which the parent of a numerical semigroup Λ is the numerical The definition of the ordinarization transform of a numerical semigroup allows the construction of a tree T*<sup>g</sup>* on the set of all numerical semigroups of a given genus rooted at the unique ordinary semigroup of this genus, where the parent of a semigroup is its ordinarization transform and the children of a semigroup are the semigroups obtained by taking away the generators one by one that are larger than the Frobenius number and adding a new non-gap smaller than the multiplicity in a licit place. To illustrate this construction with an example in Figure 2, we depicted T7.

 In [7] a new tree construction is introduced as follows. The *ordinarization transform* of a non-ordinary semigroup Λ with Frobenius number *F* and multiplicity *m* is the set Λ<sup>0</sup> = Λ \ {*m*} ∪ {*F*}. For instance, the ordinarization transform of the semigroup *H* = {0, **12**, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . } is the semigroup *H*<sup>0</sup> = {0, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, **44**, 45, 46, 47, 48, . . . } The ordinarization transform of an ordinary semigroup is then defined to be itself. Note that the genus of the ordinarization transform of

 The definition of the ordinarization transform of a numerical semigroup allows the construction of a tree T*<sup>g</sup>* on the set of all numerical semigroups of a given genus rooted at the unique ordinary semigroup of this genus, where the parent of a semigroup is its ordinarization transform and the children of a semigroup are the semigroups obtained by taking away one by one the generators that are larger than the Frobenius number and adding a new non-gap smaller than the multiplicity in a licit

 One significant difference between T*<sup>g</sup>* and T is that the first one has only a finite number of nodes. In fact, it has *n<sup>g</sup>* nodes, while T is an infinite tree. It was conjectured in [7] that the number of numerical

place. To illustrate this construction with an example in Figure 2 we depicted T7.

 obtained by adjoining to Λ its Frobenius number. For instance, the parent of the semigroup *H* = {0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . } is the semigroup *H*<sup>0</sup> = {0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, **44**, 45, 46, 47, 48, . . . }. In turn, the children of a numerical semigroup are the semigroups we obtain by taking away one by one the generators that are larger than or equal to the conductor of the semigroup. The parent of a numerical semigroup of genus *g* has prove that *ng*+<sup>1</sup> > *ng*.

**Figure 2.** The whole tree T<sup>7</sup> **Figure 2.** The whole tree T7.

 semigroups in T*<sup>g</sup>* at a given depth is at most the number of numerical semigroups in T*g*+<sup>1</sup> at the same depth. This was proved in the same reference for the lowest and largest depths. This conjecture would One significant difference between T*<sup>g</sup>* and T is that the first one has only a finite number of nodes. In fact, it has *n<sup>g</sup>* nodes, while T is an infinite tree. It was conjectured in [9] that the number of numerical semigroups in T*<sup>g</sup>* at a given depth is at most the number of numerical semigroups in T*g*+<sup>1</sup> at the same depth. This was proved in the same reference for the lowest and largest depths. This conjecture would prove that *ng*+<sup>1</sup> > *ng*.

 In Section 2 we will construct the quasi-ordinarization transform of a general semigroup, paralleling the ordinarization transform. If the quasi-ordinarization transform is applied repeatedly to a numerical semigroup, it ends up in a quasi-ordinary semigroup. In Section 3 we define the quasi-ordinarization number of a semigroup as the number of successive quasi-ordinarization transforms of the semigroup that give a quasi-ordinary semigroup. Section 4 analyzes the number of numerical semigroups of a given genus and a given quasi-ordinarization number in terms of the given parameters. We present the conjecture that the number of numerical semigroups of a given genus and a fixed quasi-ordinarization number increases with the genus and we prove it for the largest quasi-ordinarization numbers. In Section 5 we present the forest of semigroups of a given genus that is obtained when connecting each semigroup to its quasi-ordinarization transform. The forest corresponding to genus *g* is denoted F*g*. Section 6 analyzes the relationships between T, T*g*, and F*g*. In Section 2, we will construct the quasi-ordinarization transform of a general semigroup, paralleling the ordinarization transform. If the quasi-ordinarization transform is applied repeatedly to a numerical semigroup, it ends up in a quasi-ordinary semigroup. In Section 3, we define the quasi-ordinarization number of a semigroup as the number of successive quasi-ordinarization transforms of the semigroup that give a quasi-ordinary semigroup. Section 4 analyzes the number of numerical semigroups of a given genus and a given quasi-ordinarization number in terms of the given parameters. We present the conjecture that the number of numerical semigroups of a given genus and a fixed quasi-ordinarization number increases with the genus and we prove it for the largest quasiordinarization numbers. In Section 5, we present the forest of semigroups of a given genus that is obtained when connecting each semigroup to its quasi-ordinarization transform. The forest corresponding to genus *g* is denoted F*g*. Section 6 analyzes the relationships between T, T*g*, and F*g*.

 From the perspective of the forests of numerical semigroups here presented, the conjecture in Section 4 translates to the conjecture that the number of numerical semigroups in F*<sup>g</sup>* at a given depth is at most the number of numerical semigroups in F*g*+<sup>1</sup> at the same depth. The results in Section 4 provide a proof of the conjecture for the largest depths. Proving this conjecture for all depths, would prove that *ng*+<sup>1</sup> > *ng*. Hence, we expect our work to contribute to the proof of the conjectured increasingness of the sequence *n<sup>g</sup>* (A007323). From the perspective of the forests of numerical semigroups here presented, the conjecture in Section 4 translates to the conjecture that the number of numerical semigroups in F*<sup>g</sup>* at a given depth is at most the number of numerical semigroups in F*g*+<sup>1</sup> at the same depth. The results in Section 4 provide a proof of the conjecture for the largest depths. Proving this conjecture for all depths, would prove that *ng*+<sup>1</sup> > *ng*. Hence, we expect our work to contribute to the proof of the conjectured increasingness of the sequence *n<sup>g</sup>* (A007323).

#### **2. Quasi-Ordinary Semigroups and Quasi-Ordinarization Transform**

 **2. Quasi-ordinary semigroups and quasi-ordinarization transform** *Quasi-ordinary* semigroups are those semigroups for which *m* = *g* and so, there is a unique gap *Quasi-ordinary* semigroups are those semigroups for which *m* = *g* and so, there is a unique gap larger than *m*. The *sub-Frobenius number* of a non-ordinary semigroup Λ with Frobenius number *F* is the Frobenius number of Λ ∪ {*F*}.

 larger than *m*. The *sub-Frobenius number* of a non-ordinary semigroup Λ with Frobenius number *F* is the Frobenius number of Λ ∪ {*F*}. The *subconductor* of a semigroup with Frobenius number *F* is the smallest nongap in the interval The *subconductor* of a semigroup with Frobenius number *F* is the smallest nongap in the interval of nongaps immediatelly previous to *F*. For instance, the subconductor of the above example, *H* = {0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . }, is 42.

**Lemma 1.** *Let* Λ *be a non-ordinary and non quasi-ordinary semigroup, with multiplicity m, genus g, and*

*sub-Frobenius number f . Then* Λ ∪ { *f* } \ {*m*} *is another numerical semigroup of the same genus g.*

**Lemma 1.** *Let* Λ *be a non-ordinary and non quasi-ordinary semigroup, with multiplicity m, genus g, and sub-Frobenius number f . Then,* Λ ∪ { *f* } \ {*m*} *is another numerical semigroup of the same genus g.*

**Proof.** Since Λ is already a numerical semigroup, it is enough to see that *F* − *f* is not in Λ ∪ { *f* } \ {*m*}, where *F* is the Frobenius number of Λ. Notice that for a non-ordinary numerical semigroup, the difference between its Frobenius number and its sub-Frobenius number needs to be less than the multiplicity of the semigroup; hence, *F* − *f* 6∈ Λ. So, the only option for *F* − *f* to be in Λ ∪ { *f* } \ {*m*} is that *F* − *f* = *f* . In this case, any integer between 1 and *f* − 1 must be a gap, since the integers between *F* − 1 and *F* − *f* + 1 are nongaps. In this case, Λ would be quasi-ordinary, contradicting the hypotheses.

**Definition 1.** *The* quasi-ordinarization transform *of a non-ordinary and non quasi-ordinary numerical semigroup* Λ*, with multiplicity m, genus g and sub-Frobenius number f , is the numerical semigroup* Λ ∪ { *f* } \ {*m*}*.*

*The* quasi-ordinarization *of either an ordinary or quasi-ordinary semigroup is defined to be itself.*

As an example, the quasi-ordinarization of the well-tempered harmonic semigroup *H* = {0, **12**, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, . . . } used in the previous examples is *H*<sup>0</sup> = {0, 19, 24, 28, 31, 34, 36, 38, 40, **41**, 42, 43, 45, 46, 47, 48, . . . }.

**Remark 1.** *In the ordinarization and quasi-ordinarization transform process, we replace the multiplicity by the largest and second largest gap, respectively, and we obtain numerical semigroups. In general, if we replace the multiplicity by the third largest gap, we do not obtain a numerical semigroup.*

*See for instance* {0, 2, 4, 6, 8, 10, 11, . . . }*. Replacing* 2 *by* 5*, we obtain* {0, 4, 5, 6, 8, 10, 11, . . . }*, which is not a numerical semigroup since* 9 = 4 + 5 *is not in the set.*

#### **3. Quasi-Ordinarization Number**

Next, lemma explicits that there is only one quasi-ordinary semigroup with genus *g* and conductor *c* where *c* 6 2*g*.

**Lemma 2.** *For each of the positive integers g and c with c* 6 2*g, the semigroup* {0, *g*, *g* +1, . . . , *c* − 2, *c*, *c* + 1 . . . } *is the unique quasi-ordinary semigroup of genus g and conductor c.*

The quasi-ordinarization transform of a non-ordinary semigroup of genus *g* and conductor *c* can be applied subsequently and at some step, we will attain the quasiordinary semigroup of that genus and conductor, that is, the numerical semigroup {0, *g*, *g* + 1, . . . , *c* − 2, *c*, *c* + 1, . . . }. The number of such steps is defined to be the *quasi-ordinarization number* of Λ.

We denote by *\$g*,*q*, the number of numerical semigroups of genus *g* and quasiordinarization number *q*. In Table 1, one can see the values of *\$g*,*<sup>q</sup>* for genus up to 45. It has been computed by an exhaustive exploration of the semigroup tree using the RGD algorithm [12].

**Lemma 3.** *The quasi-ordinarization number of a non-ordinary numerical semigroup of genus g coincides with the number of non-zero non-gaps of the semigroup that are smaller than or equal to g* − 1*.*

**Proof.** A non-ordinary numerical semigroup of genus *g* is non-quasi-ordinary if and only if its multiplicity is at most *g* − 1. Consequently, we can repeatedly apply the quasiordinarization transform to a numerical semigroup while its multiplicity is at most *g* − 1. Furthermore, the number of consecutive transforms that we can apply before obtaining the quasi-ordinary semigroup is hence the number of its non-zero non-gaps that are at most the genus minus one.


**Table 1.** Number of semigroups of each genus and quasi-ordinarization number.

For a numerical semigroup Λ, we will consider its enumeration *λ*, that is, the unique increasing bijective map between N<sup>0</sup> and Λ. The element *λ*(*i*) is then denoted *λ<sup>i</sup>* . As a consequence of the previous lemma, for a numerical semigroup Λ with quasi-ordinarization number equal to *q*, the non-gaps that are at most *g* − 1 are exactly *λ*<sup>0</sup> = 0, *λ*1, . . . , *λq*.

**Lemma 4.** *The maximum quasi-ordinarization number of a non-ordinary semigroup of genus g is* b *g*−1 2 c*.*

**Proof.** Let Λ be a numerical semigroup with quasi-ordinarization number equal to *q*. Since the Frobenius number *F* is at most 2*g* − 1, the total number of gaps from 1 to 2*g* − 1 is *g*, and so the number of non-gaps from 1 to 2*g* − 1 is *g* − 1. The number of those non-gaps that are larger than *g* − 1 is *g* − 1 − *q*. On the other hand, *λ<sup>q</sup>* + *λ*1, *λ<sup>q</sup>* + *λ*2, . . . , 2*λ<sup>q</sup>* are different non-gaps between *g* and 2*g* − 1. So, the number of non-gaps between *g* and 2*g* − 1 is at least *q*. All these results imply that *g* − 1 − *q* > *q* and so, *q* 6 *g*−1 2 .

On the other hand, the bound stated in the lemma is attained by the hyperelliptic numerical semigroup

$$\{0, 2, 4, \dots, 2\left\lfloor \frac{g-1}{2} \right\rfloor, 2\left(\left\lfloor \frac{g-1}{2} \right\rfloor + 1\right), \dots, 2g, 2g + 1, 2g + 2, \dots\}.\tag{1}$$

We will next see that the maximum ordinarization number stated in the previous lemma is attained uniquely by the numerical semigroup in (1). To prove this result, we will need the next lemma. Let us recall that *A* + *B* = {*a* + *b* : *a* ∈ *A*, *b* ∈ *B*} and that #*A* denotes the cardinality of *A*.

**Lemma 5.** *Consider a finite subset A* = {*a*<sup>1</sup> < · · · < *an*} ⊆ N0*.*

	- *there exists a positive integer α such that a<sup>i</sup>* = *a*<sup>1</sup> + *α*(*i* − 1) *for all i with* 1 6 *i* < *n and a<sup>n</sup>* = *a*<sup>1</sup> + *nα,*
	- *there exists a positive integer α such that a<sup>i</sup>* = *a*<sup>1</sup> + *iα for all i with* 2 6 *i* 6 *n.*

**Proof.** The first item stems from the fact that if *A* = {*a*1, . . . , *an*}, then *A* + *A* must contain at least 2*a*1, *a*<sup>1</sup> + *a*2, *a*<sup>1</sup> + *a*3, . . . , *a*<sup>1</sup> + *an*, *a*<sup>2</sup> + *an*, *a*<sup>3</sup> + *an*, . . . , *an*−<sup>1</sup> + *an*, 2*an*, which are all different.

The second item easily follows from the fact that if *A* + *A* has 2*n* − 1 elements, then *A* + *A* must be exactly the set <sup>2</sup>*a*1, *a*<sup>1</sup> + *a*2, *a*<sup>1</sup> + *a*3, . . . , *a*<sup>1</sup> + *an*, *a*<sup>2</sup> + *an*, *a*<sup>3</sup> + *an*, . . . , *an*−<sup>1</sup> + *an*, 2*an*. Indeed, in this case, the increasing set {*a*<sup>1</sup> + *a*3, . . . , *a*<sup>1</sup> + *an*, *a*<sup>2</sup> + *an*, *a*<sup>3</sup> + *an*, . . . , *an*−<sup>1</sup> + *an*, 2*an*} must coincide with the increasing set {2*a*2, *a*<sup>2</sup> + *a*3, *a*<sup>2</sup> + *a*4, . . . , *a*<sup>2</sup> + *an*, *a*<sup>3</sup> + *an*, . . . , *an*−<sup>1</sup> + *an*, 2*an*}, having as a consequence that <sup>2</sup>*a*<sup>2</sup> = *a*<sup>1</sup> + *a*<sup>3</sup> and so, *a*<sup>2</sup> = *a*1+*a*<sup>3</sup> <sup>2</sup> = *a*<sup>1</sup> + *a*3−*a*<sup>1</sup> 2 , and *a*<sup>3</sup> = 2*a*<sup>2</sup> − *a*<sup>1</sup> = *a*<sup>1</sup> + 2 *a*3−*a*<sup>1</sup> 2 . Hence,

$$\begin{array}{rcl} a\_2 & = & a\_1 + \frac{a\_3 - a\_1}{2} \\ a\_3 & = & a\_1 + 2\frac{a\_3 - a\_1}{2} \end{array}$$

Similarly, one can show that 2*a*<sup>3</sup> = *a*<sup>2</sup> + *a*<sup>4</sup> and, so, *a*<sup>4</sup> = 2*a*<sup>3</sup> − *a*<sup>2</sup> = 2*a*<sup>1</sup> + 4 *a*3−*a*<sup>1</sup> 2 − *a*<sup>1</sup> − *a*3−*a*<sup>1</sup> <sup>2</sup> = *a*<sup>1</sup> + 3 *a*3−*a*<sup>1</sup> 2 . It equally follows that

$$\begin{array}{rcl} a\_4 &=& a\_1 + 3 \frac{a\_3 - a\_1}{2} \\ a\_5 &=& a\_1 + 4 \frac{a\_3 - a\_1}{2} \\ &\vdots \\ \end{array}$$

For the third item, one direction of the proof is obvious, so we just need to prove the other one, that is, if the sum contains 2*n* elements, then *a*1, . . . , *a<sup>n</sup>* must be as stated.

We will proceed by induction. Suppose that *n* = 4 and that the set *A* + *A* contains exactly 8 elements. Since the ordered sequence

$$2a\_1 < a\_1 + a\_2 < 2a\_2 < a\_2 + a\_3 < 2a\_3 < a\_3 + a\_4 < 2a\_4 \tag{2}$$

already contains 7 elements, then necessarily two of the elements *a*<sup>1</sup> + *a*3, *a*<sup>1</sup> + *a*4, *a*<sup>2</sup> + *a*<sup>4</sup> coincide with one element in (2) and the third one is not in (2). So, at least one of *a*<sup>1</sup> + *a*<sup>3</sup> and *a*<sup>2</sup> + *a*<sup>4</sup> must be in (2).

Suppose first that *a*<sup>1</sup> + *a*<sup>3</sup> is in (2). Then, necessarily *a*<sup>1</sup> + *a*<sup>3</sup> = 2*a*2, which means that *a*<sup>2</sup> − *a*<sup>1</sup> = *a*<sup>3</sup> − *a*2. Hence, there exists *α* (in fact, *α* = *a*<sup>2</sup> − *a*1) such that *a*<sup>2</sup> = *a*<sup>1</sup> + *α* and *a*<sup>3</sup> = *a*<sup>1</sup> + 2*α*. Now, the elements

$$2a\_1 < a\_1 + a\_2 < 2a\_2 < a\_2 + a\_3 < 2a\_3 \tag{3}$$

are equally separated by the same separation *α*. That is,

$$\begin{array}{rcl}(a\_1+a\_2)-(2a\_1)&=&\alpha\\(2a\_2)-(a\_1+a\_2)&=&\alpha\\(a\_2+a\_3)-(2a\_2)&=&\alpha\\(2a\_3)-(a\_2+a\_3)&=&\alpha.\end{array}$$

Additionally, the elements

$$a\_4 + a\_1 < a\_4 + a\_2 < a\_4 + a\_3 \tag{4}$$

are equally separated by the same separation *α*. That is,

$$\begin{aligned} (a\_4 + a\_3) - (a\_4 + a\_2) &= \alpha \\ (a\_4 + a\_2) - (a\_4 + a\_1) &= \alpha \end{aligned}$$

Furthermore, *A* + *A* must contain all the elements in (3) and (4) as well as the element 2*a*4, which is not in (3), nor in (4). Since #(*A* + *A*) = 8, this means that there must be exatly one element that is both in (3) and (4). The only way for this to happen is that 2*a*<sup>3</sup> = *a*<sup>4</sup> + *a*1. Consequently, *a*<sup>4</sup> + *a*<sup>1</sup> = 2*a*<sup>1</sup> + 4*α*, and so, *a*<sup>4</sup> = *a*<sup>1</sup> + 4*α*. This proves the result in the first case.

For the case in which *a*<sup>2</sup> + *a*<sup>4</sup> is in (2), it is necessary that *a*<sup>2</sup> + *a*<sup>4</sup> = 2*a*3, which means that *a*<sup>3</sup> − *a*<sup>2</sup> = *a*<sup>4</sup> − *a*3. Hence, there exists *β* (in fact, *β* = *a*<sup>3</sup> − *a*2) such that *a*<sup>3</sup> = *a*<sup>2</sup> + *β* and *a*<sup>4</sup> = *a*<sup>2</sup> + 2*β*. Now, the elements

$$2a\_2 < a\_2 + a\_3 < 2a\_3 < a\_3 + a\_4 < 2a\_4 \tag{5}$$

are equally separated by the same separation *β*. That is,

(*a*<sup>2</sup> + *a*3) − (2*a*2) = *β* (2*a*3) − (*a*<sup>2</sup> + *a*3) = *β* (*a*<sup>3</sup> + *a*4) − (2*a*3) = *β* (2*a*4) − (*a*<sup>3</sup> + *a*4) = *β*.

Additionally, the elements

$$a\_1 + a\_2 < a\_1 + a\_3 < a\_1 + a\_4 \tag{6}$$

are equally separated by the same separation *β*. That is,

$$\begin{aligned} (a\_1 + a\_3) - (a\_1 + a\_2) &= \beta \\ (a\_1 + a\_4) - (a\_1 + a\_3) &= \beta \beta \end{aligned}$$

Now, *A* + *A* must contain all the elements in (5) and (6), as well as the element 2*a*1, which is not in (5), nor in (6). Since #(*A* + *A*) = 8, this means that there must be exactly one element that is both in (5) and in (6). The only way for this to happen is that *a*<sup>1</sup> + *a*<sup>4</sup> = 2*a*2. Consequently, *a*<sup>1</sup> + *a*<sup>4</sup> = 2*a*<sup>1</sup> + 4*α*, and so, *a*<sup>4</sup> = *a*<sup>1</sup> + 4*α*. Hence, *a*<sup>2</sup> = *a*<sup>1</sup> + 2*β*, *a*<sup>3</sup> = *a*<sup>1</sup> + 3*β*, *a*<sup>4</sup> = *a*<sup>1</sup> + 4*β*. This proves the result in the second case and concludes the proof for *n* = 4.

Now, let us prove the result for a general *n* > 4. We will denote *A<sup>n</sup>* the set {*a*1, . . . , *an*}. Notice that *A*<sup>1</sup> + *A*<sup>1</sup> = {2*a*1} while, if *i* > 1, then {*ai*−<sup>1</sup> + *a<sup>i</sup>* , 2*ai*} ⊆ (*A<sup>i</sup>* + *Ai*) \ (*Ai*−<sup>1</sup> + *Ai*−1), hence, #((*A<sup>i</sup>* + *Ai*) \ (*Ai*−<sup>1</sup> + *Ai*−1)) > 2. Consequently, if #(*A<sup>n</sup>* + *An*) = 2*n*, we can affirm that there exists exactly one integer *s* such that #(*A<sup>r</sup>* + *Ar*) = 2*r* − 1, for all *r* < *s* and #(*A<sup>r</sup>* + *Ar*) = 2*r* for all *r* > *s*.

If *s* = *n*, then we already have, by the second item of the lemma, that *a<sup>i</sup>* = *a*<sup>1</sup> + (*i* − 1)*γ* for a given positive integer *γ* for all *i* < *n*.

On one hand,

$$A\_{n-1} + A\_{n-1} = \{2a\_1, 2a\_1 + \gamma, 2a\_1 + 2\gamma, 2a\_1 + 3\gamma, \dots, 2a\_1 + (2n - 4)\gamma\},\tag{7}$$

which has 2*n* − 3 elements. On the other hand,

$$A\_{n-1} + a\_{\rm n} = \{ (a\_1 + a\_{\rm n}), (a\_1 + a\_{\rm n}) + \gamma, (a\_1 + a\_{\rm n}) + 2\gamma, (a\_1 + a\_{\rm n}) + 3\gamma, \dots, (a\_1 + a\_{\rm n}) + (n - 2)\gamma \}, \tag{8}$$

has *n* − 1 elements.

Now, *A* + *A* = (*An*−<sup>1</sup> + *An*−1) ∪ (*An*−<sup>1</sup> + *an*) ∪ (2*an*). By the inclusion–exclusion principle, and since 2*a<sup>n</sup>* is not in (*An*−<sup>1</sup> + *An*−1) ∪ (*An*−<sup>1</sup> + *an*),

$$\begin{aligned} \#((A\_{n-1} + A\_{n-1}) \cap (A\_{n-1} + a\_n)) &= \#(A\_{n-1} + A\_{n-1}) + \#(A\_{n-1} + a\_n) + 1 - \#(A + A) \\ &= (2n - 3) + (n - 1) + 1 - 2n \\ &= n - 3 \end{aligned}$$

By (7) and (8), we conclude that (*a*<sup>1</sup> + *an*) + (*n* − 4)*γ* = 2*a*<sup>1</sup> + (2*n* − 4)*γ*, that is, *a<sup>n</sup>* = *a*<sup>1</sup> + *nγ*. Hence, the result follows with *α* = *γ*.

On the contrary, if *s* < *n*, then, since #(*An*−<sup>1</sup> + *An*−1) = 2(*n* − 1), we can apply the induction hypothesis and affirm that either one of the following cases, (a) or (b), holds.


$$\begin{array}{c} A\_{n-1} + A\_{n-1} = \{ 2a\_1, 2a\_1 + \alpha\_{n-1}, 2a\_1 + 2\alpha\_{n-1}, \dots \} \\ \dots, 2a\_1 + (2n - 4)\alpha\_{n-1}, 2a\_1 + (2n - 2)\alpha\_{n-1} \} \end{array}$$

and

$$A\_{n-1} + a\_n = \begin{Bmatrix} (a\_1 + a\_n)\_\prime (a\_1 + a\_n) + a\_{n-1} \, (a\_1 + a\_n) + 2a\_{n-1}, \dots \\ \dots, (a\_1 + a\_n) + (n-3)a\_{n-1} \, (a\_1 + a\_n) + (n-1)a\_{n-1} \end{Bmatrix}$$

In case (b), we will have

$$\begin{array}{c} A\_{n-1} + A\_{n-1} = \{ 2a\_1, 2a\_1 + 2a\_{n-1}, 2a\_1 + 3a\_{n-1}, \dots \\ \dots, 2a\_1 + (2n - 3)a\_{n-1}, 2a\_1 + (2n - 2)a\_{n-1} \} \end{array}$$

and

$$A\_{n-1} + a\_n = \{ (a\_1 + a\_n), (a\_1 + a\_n) + 2a\_{n-1}, (a\_1 + a\_n) + 3a\_{n-1}, \dots \} $$

Now,

$$\begin{aligned} \#((A\_{n-1} + A\_{n-1}) \cap (A\_{n-1} + a\_n)) &= \#(A\_{n-1} + A\_{n-1}) + \#(A\_{n-1} + a\_n) + 1 - \#(A + A) \\ &= \#(A\_{n-1} + A\_{n-1}) - n \\ &= n - 2. \end{aligned}$$

This is only possible in case (b) with

$$\begin{aligned} \left( \left( A\_{n-1} + A\_{n-1} \right) \cap \left( A\_{n-1} + a\_n \right) \right) &= \left\{ \left( a\_1 + a\_2 \right), \left( a\_1 + a\_n \right) + 2a\_{n-1}, \left( a\_1 + a\_n \right) + 3a\_{n-1}, \dots \right\} \\ &\dots, \left( a\_1 + a\_n \right) + (n - 2)a\_{n-1} \right\} \end{aligned}$$

and, hence, with (*a*<sup>1</sup> + *an*) + (*n* − 2)*αn*−<sup>1</sup> = 2*a*<sup>1</sup> + (2*n* − 2)*αn*−1, that is, *a<sup>n</sup>* = *a*<sup>1</sup> + *nαn*−1, hence yielding the result with *α* = *αn*−1.

**Lemma 6.** *Let g* > 2 *and g* 6= 4, *g* 6= 6*. The unique non-quasi-ordinary numerical semigroup of genus g and quasi-ordinarization number* b *g*−1 2 c *is* {0, 2, 4, . . . , 2*g*, 2*g* + 2, 2*g* + 3 . . . }*.*

**Proof.** If *g* = 3, there is only one numerical semigroup non-ordinary and non-quasiordinary as we can observe in Figure 1, and it is exactly {0, 2, 4, 6, . . . }, which indeed, has a quasi-ordinarization number b *g*−1 2 c and it is of the form {0, 2, 4, . . . , 2*g*, 2*g* + 1, 2*g* + 2, . . . }. The case *g* = 4 and *g* = 6 are excluded from the statement (and analyzed in Remark 2). So, we can assume that either *g* = 5 or *g* > 6.

Suppose that the quasi-ordinarization number of Λ is b *g*−1 2 c. Since *λ* b *g*−1 2 c 6 *g* − 1, we know that the set of all non-gaps between 0 and 2*g* − 2 must contain all the sums

$$
\Sigma = \{\lambda\_i + \lambda\_j : 0 \le i, j \ll \lfloor \frac{\mathbf{g} - 1}{2} \rfloor\}.
$$

However, the number of non-gaps between 0 and 2*g* − 2 is either *g* − 1 or *g* depending on whether 2*g* − 1 is a gap or not. So, #Σ 6 *g*. On the other hand, by Lemma 5, #Σ > 2b *g*−1 2 c + 1.

If *g* is odd, we get that 2b *g*−1 2 c + 1 = *g* and so, #Σ = *g*. Then, by the second item in Lemma 5, we get that *λ<sup>i</sup>* = *iλ*<sup>1</sup> for *i* 6 *g*−1 2 . Now, *λ <sup>g</sup>*−<sup>1</sup> 2 = *g*−1 2 *λ*<sup>1</sup> and, since *λ <sup>g</sup>*−<sup>1</sup> 2 6 *g* − 1, one can deduce that *λ*<sup>1</sup> 6 2. If *λ*<sup>1</sup> = 1 this contradicts *g* > 1. So, *λ<sup>i</sup>* = 2*i* for 0 6 *i* 6 *g*−1 2 and the remaining non-gaps between *g* and 2*g* are necessarily *λ<sup>i</sup>* = 2*i* for *i* = *g*−1 <sup>2</sup> + 1 to *i* = *g*.

If *g* is even, then *g* − 1 6 #Σ 6 *g*. If #Σ = *g*, then, since the number of summands in the sum Σ is at least 4 (because we excluded the even genera 4 and 6), we can apply the third item in Lemma 5. Then, we obtain *λ <sup>g</sup>* <sup>2</sup> <sup>−</sup><sup>1</sup> <sup>=</sup> *g* 2 *λ*1. This, together with *λ <sup>g</sup>* <sup>2</sup> <sup>−</sup><sup>1</sup> <sup>6</sup> *<sup>g</sup>* <sup>−</sup> <sup>1</sup> implies that *λ*<sup>1</sup> 6 2 *g*−1 *<sup>g</sup>* < 2. So, *λ*<sup>1</sup> = 1, contradicting *g* > 1. Hence, it must be Σ = *g* − 1. If Σ = *g* − 1, then, by the second item in Lemma 5, we obtain *λ<sup>i</sup>* = *iλ*<sup>1</sup> for all *i* 6 *g* <sup>2</sup> − 1. Now, *λ <sup>g</sup>* <sup>2</sup> <sup>−</sup><sup>1</sup> = ( *<sup>g</sup>* <sup>2</sup> − 1)*λ*<sup>1</sup> and, since *λ <sup>g</sup>* <sup>2</sup> <sup>−</sup><sup>1</sup> <sup>6</sup> *<sup>g</sup>* <sup>−</sup> 1, one can deduce that *<sup>λ</sup>*<sup>1</sup> <sup>6</sup> <sup>2</sup> *g*−1 *g*−2 . However, 2 *g*−1 *<sup>g</sup>*−<sup>2</sup> <sup>&</sup>lt; <sup>3</sup> if *<sup>g</sup>* <sup>&</sup>gt; 5. So, *<sup>λ</sup>*<sup>1</sup> con only be <sup>1</sup> or 2. If *<sup>λ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> this contradicts *<sup>g</sup>* <sup>&</sup>gt; 1. So, *<sup>λ</sup><sup>i</sup>* <sup>=</sup> <sup>2</sup>*<sup>i</sup>* for 0 6 *i* 6 *g* <sup>2</sup> − 1 and the remaining non-gaps between *g* and 2*g* are necessarily *λ<sup>i</sup>* = 2*i* for *i* = *g* 2 to *i* = *g*.

**Remark 2.** *For g* = 4*, the maximum quasi-ordinarization number* b *g*−1 2 c = 1 *is, in fact, attained by three of the 7 semigroups of genus 4. The semigroups whose quasi-ordinarization number is maximum are* {0, 3, 6, . . . }*,* {0, 2, 4, 6, 8, . . . }*,* {0, 3, 5, 6, 8, . . . }*.*

*For g* = 6*, the maximum quasi-ordinarization number* b *g*−1 2 c = 2 *is, in fact, attained by two of the 23 semigroups of genus 6. The semigroups whose quasi-ordinarization number is maximum are* {0, 2, 4, 6, 8, 10, 12, . . . }*,* {0, 4, 5, 8, 9, 10, 12, . . . }*.*

*Hence, g* = 4 *and g* = 6 *are exceptional cases.*

#### **4. Analysis of** *\$g***,***<sup>q</sup>*

Let us denote *og*,*<sup>r</sup>* , the number of numerical semigroups of genus *g* and ordinarization number *r* and *\$g*,*q*, the number of numerical semigroups of genus *g* and quasiordinarization number *r*.

We can observe a behavior of *\$g*,*<sup>q</sup>* very similar to the behavior of *og*,*<sup>r</sup>* introduced in [9]. Indeed, for odd *g* and large *r*, it holds *\$g*,*<sup>q</sup>* = *og*,*<sup>r</sup>* and for even *g* and large *q*, it holds *\$g*,*<sup>q</sup>* = *og*,*r*+1. We will give a partial proof of these equalities at the end of this section.

It is conjcetured in [9] that, for each genus *g* ∈ N<sup>0</sup> and each ordinarization number *r* ∈ N0,

*og*,*<sup>r</sup>* 6 *og*+1,*<sup>r</sup>* .

We can write the new conjecture below paralleling this.

**Conjecture 1.** *For each genus g* ∈ N<sup>0</sup> *and each quasi-ordinarization number q* ∈ N0*,*

*\$g*,*<sup>q</sup>* 6 *\$g*+1,*<sup>q</sup>* .

Now, we will provide some results on *\$g*,*<sup>q</sup>* for high quasi-ordinarization numbers. First, we will need Fre˘ıman's Theorem [25,26] as formulated in [27].

**Theorem 2** (Fre˘ıman)**.** *Let A be a set of integers such that* #*A* = *k* > 3*. If* #(*A* + *A*) 6 3*k* − 4*, then A is a subset of an arithmetic progression of length* #(*A* + *A*) − *k* + 1 6 2*k* − 3*.*

The next lemma is a consequence of Fre˘ıman's Theorem. The lemma shows that the first non-gaps of numerical semigroups of large quasi-ordinarization number must be even.

**Lemma 7.** *If a semigroup* Λ *of genus g has quasi-ordinarization number q with <sup>g</sup>*+<sup>1</sup> <sup>3</sup> 6 *q* 6 b *g*−1 2 c *then all its non-gaps which are less than or equal to g* − 1 *are even.*

**Proof.** Suppose that Λ is a semigroup of genus *g* and quasi-ordinarization number *q* > *g*+1 3 .

Since the quasi-ordinarization is *q*, this means that *λ*<sup>0</sup> = 0, *λ*1, . . . , *λ<sup>q</sup>* 6 *g* − 1 and *λq*+<sup>1</sup> > *g*, hence Λ ∩ [0, *g* − 1] = {*λ*0, *λ*1, . . . , *λq*}. Let *A* = Λ ∩ [0, *g* − 1]. By the previous equality, #*A* = *q* + 1. We have that the elements in *A* + *A* are upper bounded by 2*g* − 2 and so *A* + *A* ⊆ Λ ∩ [0, 2*g* − 2]. Then, #(*A* + *A*) 6 #(Λ ∩ [0, 2*g* − 2]) < #(Λ ∩ [0, 2*g*]). Since the Frobenius number of Λ is at most 2*g* − 1, #(Λ ∩ [0, 2*g*]) = *g* + 1 and, so, #(*A* + *A*) 6 *g*. Now, since *q* > *g*+1 3 , we have *g* 6 3*q* − 1 = 3(*q* + 1) − 4 = 3(#*A*) − 4 and we can apply Theorem 2 with *k* = *q* + 1. Thus, we have that *A* is a subset of an arithmetic progression of length at most *g* − *k* + 1 = *g* − *q*.

Let *d*(*A*) be the difference between consecutive terms of this arithmetic progression. The number *d*(*A*) can not be larger than or equal to three since otherwise *λ<sup>q</sup>* > *q* · *d*(*A*) > 3*q* > 3 *g*+1 <sup>3</sup> > *g*, a contradiction with *q* being the quasi-ordinarization number.

If *d*(*A*) = 1, then *A* ⊆ [0, *g* − *q* − 1] and so Λ ∩ [*g* − *q*, *g* − 1] = ∅. We claim that in this case *A* ⊆ {0} ∪ [d *g* 2 e, *g* − *q* − 1]. Indeed, suppose that *x* ∈ *A*. Then, 2*x* satisfies either 2*x* 6 *g* − *q* − 1 or 2*x* > *g*. If the second inequality is satisfied, then it is obvious that *x* ∈ {0} ∪ [d *g* 2 e, *g* − *q* − 1]. If the first inequality is satisfied, then we will prove that *mx* 6 *g* − *q* − 1 for all *m* > 2 by induction on *m* and this leads to *x* = 0. Indeed, if *mx* 6 *g* − *q* − 1, then *x* 6 *g*−*q*−1 *<sup>m</sup>* 6 *g*− *g*+1 <sup>3</sup> −1 *<sup>m</sup>* = 2*g*−4 <sup>3</sup>*<sup>m</sup>* < 2*g* 3*m* . Now, (*m* + 1)*x* < 2*g*(*m*+1) <sup>3</sup>*<sup>m</sup>* = (2*m*+2)*g* 3*m* and since *m* > 2, we have (*m* + 1)*x* < (2*m*+*m*)*g* <sup>3</sup>*<sup>m</sup>* = *g* and so (*m* + 1)*x* 6 *g* − 1. Since (*m* + 1)*x* is in Λ ∩ [0, *g* − 1] = *A* ⊆ [0, *g* − *q* − 1], this means that (*m* + 1)*x* 6 *g* − *q* − 1 and this proves the claim.

Now, *A* ⊆ {0} ∪ [d *g* 2 e, *g* − *q* − 1] together with #*A* = *q* + 1 implies that *q* 6 *g* − *q* − d *g* 2 e = b *g* 2 c − *q* 6 *g* 2 − *g*+1 <sup>3</sup> = *g*−2 <sup>6</sup> < *q*, a contradiction.

So, we deduce that *d*(*A*) = 2, leading to the proof of the lemma.

The next lemma was proved in [9].

**Lemma 8.** *Suppose that a numerical semigroup* Λ *has ω gaps between* 1 *and n* − 1 *and n* > 2*ω* + 2*, then*


Let Λ be a numerical semigroup. As in [9], let us say that a set *B* ⊂ N<sup>0</sup> is Λ-*closed* if for any *b* ∈ *B* and any *λ* in Λ, the sum *b* + *λ* is either in *B* or it is larger than max(*B*). If *B* is Λ-closed, so is *B* − min(*B*). Indeed, *b* − min(*B*) + *λ* = (*b* + *λ*) − min(*B*) is either in *B* − min(*B*) or it is larger than max(*B*) − min(*B*) = max(*B* − min(*B*)). The new Λ-closed set contains 0. We will denote by *C*(Λ, *i*), the Λ-closed sets of size *i* that contain 0.

Let S*<sup>γ</sup>* be the set of numerical semigroups of genus *γ*. It was proved in [9] that, for *r*, an integer with *<sup>g</sup>*+<sup>2</sup> <sup>3</sup> 6 *r* 6 b *g* 2 c, it holds

$$o\_{\mathbb{S}^{r}} = \sum\_{\Omega \in \mathcal{S}\_{(\lfloor \frac{\mathbb{S}}{2} \rfloor - r)}} \#\mathbb{C}(\Omega, \left\lfloor \frac{\mathbb{S}}{2} \right\rfloor - r + 1).$$

We will see now that, for *q* an integer with *<sup>g</sup>*+<sup>1</sup> <sup>3</sup> 6 *q* 6 b *g*−1 2 c, it holds

$$\varrho\_{\mathcal{S},\mathcal{q}} = \sum\_{\Omega \in \mathcal{S}\_{(\lfloor \frac{\mathfrak{S}-1}{2} \rfloor - q)}} \#\mathbb{C}(\Omega, \left\lfloor \frac{\mathfrak{g}-1}{2} \right\rfloor - q + 1).$$

This proves that, for *q*, an integer with *<sup>g</sup>*+<sup>2</sup> <sup>3</sup> 6 *q* 6 b *g*−1 2 c, we have

$$\rho\_{\mathcal{G}\mathcal{A}} = \begin{cases} \begin{array}{cc} o\_{\mathcal{G},\emptyset} & \text{if } \mathcal{g} \text{ is odd,} \\ o\_{\mathcal{G},\emptyset+1} & \text{if } \mathcal{g} \text{ is even.} \end{array} \end{cases}$$

**Theorem 3.** *Let <sup>g</sup>* <sup>∈</sup> <sup>N</sup>0*, <sup>g</sup>* <sup>&</sup>gt; <sup>1</sup>*, and let <sup>q</sup> be an integer with <sup>g</sup>*+<sup>1</sup> <sup>3</sup> 6 *q* 6 b *g*−1 2 c*. Define ω* = b *g*−1 2 c − *q*

*1. If* Ω *is a numerical semigroup of genus ω and B is an* Ω*-closed set of size ω* + 1 *and first element equal to* 0 *then*

$$\{2j: j \in \Omega\} \cup \{2j-2\max(B)+2\mathbf{g}+1: j \in B\} \cup (\mathbf{2g}+\mathbb{N}\_0)$$

*is a numerical semigroup of genus g and quasi-ordinarization number equal to q.*

*2. All numerical semigroups of genus g and quasi-ordinarization number q can be uniquely written as*

$$\{2j: j \in \Omega\} \cup \{2j-2\max(B)+2\mathbf{g}+\mathbf{1}: j \in B\} \cup \left(2\mathbf{g}+\mathbb{N}\_0\right)$$

*for a unique numerical semigroup* Ω *of genus ω and a unique* Ω*-closed set B of size ω* + 1 *and first element equal to* 0*.*

*3. The number ρg*,*<sup>q</sup> of numerical semigroups of genus g and quasi-ordinarization number q depends only on ω. It is exactly*

$$\sum\_{\Omega \in \mathcal{S}\_{\omega}} \#\mathcal{C}(\Omega, \omega + 1).$$

#### **Proof.**

1. Suppose that Ω is a numerical semigroup of genus *ω* and *B* is an Ω-closed set of size *ω* + 1 and first element equal to 0. Let *X* = {2*j* : *j* ∈ Ω}, *Y* = {2*j* − 2 max(*B*) + 2*g* + 1 : *j* ∈ *B*}, and *Z* = (2*g* + N0).

First of all, let us see that the complement N<sup>0</sup> \ (*X* ∪ *Y* ∪ *Z*) has *g* elements. Notice that all elements in *X* are even while all elements in *Y* are odd. So, *X* and *Y* do not intersect. Additionally, the unique element in *Y* ∩ *Z* is 2*g* + 1. The number of elements in the complement will be

$$\begin{aligned} \mathsf{\#N}\_0 \mid (X \cup Y \cup Z) &= \mathsf{\mathbb{Z}} \mathsf{\mathbb{-}} \#\{x \in X : x < 2\mathsf{g}\} - \mathsf{\#Y} + 1 \\ &= \mathsf{\mathbb{Z}} \mathsf{\mathbb{-}} \mathsf{\mathbb{-}} \{s \in \Omega : s < \mathsf{g}\} - \mathsf{\#B} + 1 \\ &= \mathsf{\mathbb{Z}} \mathsf{\mathbb{-}} \mathsf{\mathbb{-}} \omega - \mathsf{\#\{s \in \Omega : s < \mathsf{g}\}}. \end{aligned}$$

We know that all gaps of Ω are at most 2*ω* − 1 = 2(b *g*−1 2 c − *q*) − 1 6 *g* − 1 − 2*q* − 1 < *g*. So, #{*s* ∈ Ω : *s* < *g*} = *g* − *ω* and we conclude that #N<sup>0</sup> \ (*X* ∪ *Y* ∪ *Z*) = *g*.

Before proving that *X* ∪ *Y* ∪ *Z* is a numerical semigroup, let us prove that the number of non-zero elements in *X* ∪ *Y* ∪ *Z*, which are smaller than or equal to *g* − 1 is *q*. Once we prove that *X* ∪ *Y* ∪ *Z* is a numerical semigroup, this will mean, by Lemma 3, that it has quasi-ordinarization number *q*. On the one hand, all elements in *Y* are larger than *g* − 1. Indeed, if *λ* is the enumeration of Ω (i.e., Ω = {*λ*0, *λ*1, . . . } with *λ<sup>i</sup>* < *λi*+<sup>1</sup> ), then max(*B*) 6 *λ<sup>ω</sup>* 6 2*ω* = 2b *g*−1 2 c − 2*q* 6 *g* − 1 − 2 *g*+1 <sup>3</sup> < *g* 3 . Now, for any *j* ∈ *B*, 2*j* − 2 max(*B*) + 2*g* + 1 > 2*g* − 2 max(*B*) > *g*. On the other hand, all gaps of Ω are at most 2*ω* − 1 = 2b *g*−1 2 c − 2*q* − 1 < *g* − 2(*g*+1) <sup>3</sup> − 1 < *g* <sup>3</sup> − 1 and so all the even integers not belonging to *X* are less than *g*. So, the number of non-zero non-gaps of *X* ∪ *Y* ∪ *Z* smaller than or equal to *g* − 1 is b *g*−1 2 c − *ω* = *q*.

To see that *X* ∪ *Y* ∪ *Z* is a numerical semigroup, we only need to see that it is closed under addition. It is obvious that *X* + *Z* ⊆ *Z*, *Y* + *Z* ⊆ *Z*, *Z* + *Z* ⊆ *Z*. It is also obvious that *X* + *X* ⊆ *X* since Ω is a numerical semigroup and that *Y* + *Y* ⊆ *Z* since, as we proved before, all elements in *Y* are larger than *g*.

It remains to see that *X* + *Y* ⊆ *X* ∪ *Y* ∪ *Z*. Suppose that *x* ∈ *X* and *y* ∈ *Y*. Then, *x* = 2*i* for some *i* ∈ Ω and *y* = 2*j* − 2 max(*B*) + 2*g* + 1 for some *j* ∈ *B*. Then, *x* + *y* = 2(*i* + *j*) − 2 max(*B*) + 2*g* + 1. Since *B* is Ω-closed, we have that either *i* + *j* ∈ *B* and so *x* + *y* ∈ *Y* or *i* + *j* > max(*B*). In this case, *x* + *y* ∈ *Z*. So, *X* + *Y* ⊆ *Y* ∪ *Z*.

2. First of all notice that, since the Frobenius number of a semigroup Λ of genus *g* is smaller than 2*g*, it holds

$$
\Lambda \cap (\mathfrak{Z}\mathfrak{g} + \mathbb{N}\_0) = (\mathfrak{Z}\mathfrak{g} + \mathbb{N}\_0).
$$

For any numerical semigroup Λ, the set Ω = { *j* 2 : *j* ∈ Λ ∩ (2N0)} is a numerical semigroup. If Λ has a quazi-ordinarization number *q* > *g*+1 3 then, by Lemma 7,

$$
\Lambda \cap [0, \emptyset - 1] = (2\Omega) \cap [0, \emptyset - 1].
$$

The semigroup Ω has exactly *q* + 1 non-gaps between 0 and b *g*−1 2 c and *ω* = b *g*−1 2 c − *q* gaps between 0 and b *g*−1 2 c. We can use Lemma 8 with *n* = b *g*+1 2 c since

$$2\omega + 2 = 2\left\lfloor \frac{g-1}{2} \right\rfloor - 2q + 2 \lesssim g - 1 - \frac{2(g+1)}{3} + 2 = \frac{g+1}{3}$$

,

which implies 2*ω* + 2 6 *g*+1 <sup>3</sup> 6 b *g*+1 2 c = *n*. Then, the genus of Ω is *ω* and the Frobenius number of Ω is at most b *g*+1 2 c. This means that all even integers larger than *g* − 1 belong to Λ.

Define *D* = (Λ ∩ [0, 2*g*]) \ 2Ω. That is, *D* is the set of odd non-gaps of Λ smaller than <sup>2</sup>*g*. We claim that *<sup>B</sup>*¯ <sup>=</sup> { *j*−1 2 : *j* ∈ *D* ∪ {2*g* + 1}} is a Ω-closed set of size *ω* + 1. The size follows from the fact that the number of non-gaps of Λ between *g* and 2*g* is *g* − *q* and that the number of even integers in the same interval is d *g*+1 2 e. Suppose that *<sup>λ</sup>* <sup>∈</sup> <sup>Ω</sup> and *<sup>b</sup>* <sup>∈</sup> *<sup>B</sup>*¯. Then, *<sup>b</sup>* <sup>=</sup> *j*−1 2 for some *j* in *D* ∪ {2*g* + 1} and *b* + *λ* = (*j*+2*λ*)−1 2 . If (*j*+2*λ*)−<sup>1</sup> <sup>2</sup> <sup>&</sup>gt; max(*B*¯) = (2*g*+1)−<sup>1</sup> 2 , we are done. Otherwise, we have *j* + 2*λ* 6 2*g*. Since Λ is a numerical semigroup and both *j*, 2*λ* ∈ Λ, it holds *j* + 2*λ* ∈ Λ ∩ [0, 2*g*]. Furthermore, *j* + 2*λ* is odd since *j* is also. So, *b* + *λ* is either larger than max(*B*¯) or it is in *<sup>B</sup>*¯. Then, *<sup>B</sup>* <sup>=</sup> *<sup>B</sup>*¯ <sup>−</sup> min(*B*¯) is a <sup>Λ</sup>-closed set of size *<sup>ω</sup>* <sup>+</sup> 1 and first element zero.

3. The previous two points define a bijection between the set of numerical semigroups in S*<sup>g</sup>* of quasi-ordinarization number *q* and the set {*C*(Ω, *ω* + 1) : Ω ∈ S*ω*}. Hence, *<sup>ρ</sup>g*,*<sup>q</sup>* = <sup>∑</sup>Ω∈S*<sup>ω</sup>* #*C*(Ω, *ω* + 1).

$$\square$$

**Corollary 1.** *Suppose that <sup>g</sup>*+<sup>2</sup> <sup>3</sup> 6 *q* 6 b *g*−1 2 c*. Then,*

$$\rho\_{\mathcal{G}\mathcal{A}} = \begin{cases} \begin{array}{c} o\_{\mathcal{S},\emptyset} \\ o\_{\mathcal{S},\emptyset+1} \end{array} & \begin{array}{c} \text{if } \mathcal{g} \text{ is odd,} \\ \text{if } \mathcal{g} \text{ is even.} \end{array} \end{cases}$$

Define, as in [9], the sequence *<sup>f</sup><sup>ω</sup>* by *<sup>f</sup><sup>ω</sup>* = <sup>∑</sup>Ω∈S*<sup>ω</sup>* #*C*(Ω, *ω* + 1). The first elements in the sequence, from *f*<sup>0</sup> to *f*<sup>15</sup> are


We remark that this sequence appears in [5], where Bernardini and Torres proved that the number of numerical semigroups of genus 3*ω* whose number of even gaps is *ω* is exactly *fω*. It corresponds to the entry A210581 of The On-Line Encyclopedia of Integer Sequences [23].

We can deduce the values *\$g*,*<sup>q</sup>* using the values in the previous table together with Theorem 3 for any *g*, whenever *q* > max( *g*+2 3 , b *g*−1 2 c − 15).

The next corollary is a consequence of the fact that the sequence *f<sup>ω</sup>* is increasing for *ω* between 0 and 15.

**Corollary 2.** *For any g* ∈ N *and any q* > max( *g* <sup>3</sup> + 1, b *g* 2 c − 15)*, it holds \$g*,*<sup>q</sup>* > *\$g*+1,*<sup>q</sup> .*

If we proved that *f<sup>ω</sup>* 6 *fω*+<sup>1</sup> for any *ω*, this would imply *\$g*,*<sup>q</sup>* 6 *\$g*+1,*<sup>q</sup>* for any *q* > *g* 3 .

#### **5. The Forest** F*<sup>g</sup>*

Fix a genus *g*. We can define a graph in which the nodes are all semigroups of that genus and whose edges connect each semigroup to its quasi-ordinarization transform, if it is not itself. The graph is a forest F*<sup>g</sup>* rooted at all ordinary and quasi-ordinary semigroups of genus *g*. In particular, the quasi-ordinarization transform defines, for each fixed genus and conductor, a tree rooted at the unique quasi-ordinary semigroup of that genus and conductor, given in Lemma 2. See F<sup>4</sup> in Figure 3, F<sup>6</sup> in Figure 4, and F<sup>7</sup> in Figure 5.

In the forest F*g*, we know that the parent of a numerical semigroup that is not a root is its quasi-ordinarization transform. Let us analyze now, what the children of a numerical semigroup are. The next result is well known and can be found, for instance, in [2]. We use Λ∗ to denote Λ \ {0}.

**Lemma 9.** *Suppose that* Λ *is a numerical semigroup and that a* ∈ N<sup>0</sup> \ Λ*. The set a* ∪ Λ *is a numerical semigroup if and only if*


and 15.

**5. The forest** F*<sup>g</sup>*

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

Figure 4, and F<sup>7</sup> in Figure 5.

any *g*, whenever *q* > max(

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

1 2 3 4 5 6 7 8 9 10 11 12 13 14

c − 15).

**Corollary 2.** *For any g* ∈ N *and any q* > max(

• 3*a* ∈ Λ*.*

The elements *a* ∈ N<sup>0</sup> \ Λ such that *a* + Λ ⊆ Λ, are denoted *pseudo-Frobenius numbers* of Λ. The elements *a* ∈ N<sup>0</sup> \ Λ such that {2*a*, 3*a*} ⊆ Λ, are denoted *fundamental gaps* of Λ. The elements satisfying the three conditions will be called *candidates*.

Suppose that a numerical semigroup Λ with Frobenius number *F* has children in F*g*. Let *e*1, . . . ,*e<sup>r</sup>* be the generators of Λ between the subconductor and *F* − 1. For *i* = 1, . . . ,*r*, let *c i* , . . . , *c i ki* be the candidates of Λ \ {*ei*}. The children of Λ in F*<sup>g</sup>* are the semigroups of the form Λ \ {*ei*} ∪ {*c i j* }, for *i* = 1, . . . ,*r* and *j* = 1, . . . , *k<sup>i</sup>* . Version June 3, 2021 submitted to *Journal Not Specified* 13 of 17


 We remark that this sequence appears in [1], where Bernardini and Torres proved that the number **Figure 4.** F<sup>6</sup> **Figure 4.** F6.

of numerical semigroups of genus 3*ω* whose number of even gaps is *ω* is exactly *fω*. It corresponds to

We can deduce the values *\$g*,*<sup>q</sup>* using the values in the previous table together with Theorem 3 for

The next corollary is a consequence of the fact that the sequence *f<sup>ω</sup>* is increasing for *ω* between 0

+ 1, b

 Fix a genus *g*. We can define a graph in which the nodes are all semigroups of that genus and whose edges connect each semigroup to its quasi-ordinarization transform, if it is not itself. The graph is a forest F*<sup>g</sup>* rooted at all ordinary and quasi-ordinary semigroups of genus *g*. In particular, the quasi-ordinarization transform defines, for each fixed genus and conductor, a tree rooted at the unique quasi-ordinary semigroup of that genus and conductor, given in Lemma 2. See F<sup>4</sup> in Figure 3, F<sup>6</sup> in

 In the forest F*<sup>g</sup>* we know that the parent of a numerical semigroup that is not a root is its quasi-ordinarization transform. Let us analyze now, what are the children of a numerical semigroup.

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

to denote Λ \ {0}.

**Figure 5.** F<sup>7</sup>

*g* 

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

c − 15)*, it holds \$g*,*<sup>q</sup>* > *\$g*+1,*<sup>q</sup>*

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

for any *q* >

*g* 

*g* 

*.*

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

If we proved that *f<sup>ω</sup>* 6 *fω*+<sup>1</sup> for any *ω*, this would imply *\$g*,*<sup>q</sup>* 6 *\$g*+1,*<sup>q</sup>*

The next result is well know and can be found, for instance, in [25]. We use Λ∗

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> . . .

.

the entry A210581 of The On-Line Encyclopedia of Integer Sequences [23].

*g*+2 , b *g*−1 

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

## 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 2 3 4 5 6 7 8 9 10 11 12

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

**Figure 4.** F<sup>6</sup>

**Figure 5.** F<sup>7</sup> **Figure 5.** F7.

#### **6. Relating** F*g***,** T*g***, and** T

Now, we analyze the relation between the kinship of different nodes in F*g*, T*g*, and T. If two semigroups are children of the same semigroup Λ, then they are called *siblings*. If Λ<sup>1</sup> and Λ<sup>2</sup> are siblings, and Λ<sup>3</sup> is a child of Λ2, then we say that Λ<sup>3</sup> is a *niece*/*nephew* of Λ1.

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . . <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . . <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . . <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> . . .

Let *q*(Λ) denote the quasi-ordinarization of Λ. The next lemmas are quite immediate from the definitions.

**Lemma 10.** *If* Λ<sup>1</sup> *is a child of* Λ<sup>2</sup> *in* T*, then q*(Λ1) *is a niece/nephew of q*(Λ2) *in* T*.*

As an example, Λ<sup>1</sup> = {0, 4, 5, 8, 9, 10, 12, . . . } is a child of Λ<sup>2</sup> = {0, 4, 5, 8, . . . } in T, while *q*(Λ1) = {0, 5, 7, 8, 9, 10, 12, . . . } is a niece of *q*(Λ2) = {0, 5, 6, 8, . . . } in T.

**Lemma 11.** *If* Λ<sup>1</sup> *and* Λ<sup>2</sup> *are siblings in* T*, then they are siblings in* T*<sup>g</sup> but not in* F*g.*

As an example, Λ<sup>1</sup> = {0, 5, 7, 9, 10, 11, 12, 14, . . . } and Λ<sup>2</sup> = {0, 5, 7, 9, 10, 12, . . . } are siblings in T and in T<sup>7</sup> (see Figure 2), but they are not siblings in F<sup>7</sup> (see Figure 5).

**Lemma 12.** *If* Λ<sup>1</sup> *and* Λ<sup>2</sup> *are siblings in* T*g, then q*(Λ1) *and q*(Λ2) *are siblings in* T*.*

As an example, Λ<sup>1</sup> = {0, 3, 6, 9, 10, 12, . . . } and Λ<sup>2</sup> = {0, 5, 6, 10, . . . } are siblings in T<sup>7</sup> (see Figure 2), and *q*(Λ1) = {0, 6, 8, 9, 10, 12, . . . } and *q*(Λ2) = {0, 6, 8, 10, . . . } are siblings in T.

As a consequence of the previous two lemmas, we obtain this final lemma.

**Lemma 13.** *If* Λ<sup>1</sup> *and* Λ<sup>2</sup> *are siblings in* T*, then q*(Λ1) *and q*(Λ2) *are siblings in* T*.*

As an example, Λ<sup>1</sup> = {0, 5, 7, 9, 10, 11, 12, 14, . . . } and Λ<sup>2</sup> = {0, 5, 7, 9, 10, 12, . . . } are siblings in T and *q*(Λ1) = {0, 7, 8, 9, 10, 11, 12, 14, . . . } and *q*(Λ2) = {0, 7, 8, 9, 10, 12, . . . } are siblings in T.

#### **7. Conclusions**

Quasi-ordinary semigroups are those semigroups that have all gaps except one in a row, while ordinary semigroups have all gaps in a row.

We defined a quasi-ordinarization transform that, applied repeatedly to a non-ordinary numerical semigroup stabilizes in a quasi-ordinary semigroup of the same genus.

From this transform, fixing a genus *g*, we can define a forest F*<sup>g</sup>* whose nodes are all semigroups of genus *g*, whose roots are all ordinary and quasi-ordinary semigroups of that genus, and whose edges connect each non-ordinary and non-quasi-ordinary numerical semigroup to its quasi-ordinarization transform.

We conjectured that the number of numerical semigroups in F*<sup>g</sup>* at a given depth is at most the number of numerical semigroups in F*g*+<sup>1</sup> at the same depth. We provided a proof of the conjecture for the largest possible depths. Proving this conjecture for all depths would prove the conjecture that *ng*+<sup>1</sup> > *ng*. Hence, we expect our work to be a step toward the proof of the conjectured increasingness of the sequence *ng*.

**Author Contributions:** All authors contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partly supported by the Catalan Government under grant 2017 SGR 00705 and by the Spanish Ministry of Economy and Competitivity under grant TIN2016-80250-R.

**Institutional Review Board Statement:** Not Applicable.

**Informed Consent Statement:** Not Applicable.

**Data Availability Statement:** Not Applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Cryptanalysis of a Group Key Establishment Protocol**

**Jorge Martínez Carracedo 1,† and Adriana Suárez Corona 2,\* ,†**


**Abstract:** In this paper, we analyze the security of a group key establishment scheme proposed by López-Ramos et al. This proposal aims at allowing a group of users to agree on a common key. We present several attacks against the security of the proposed protocol. In particular, an active attack is presented, and it is also proved that the protocol does not provide forward secrecy.

**Keywords:** cryptanalysis; group key establishment

#### **1. Introduction**

**Citation:** Carracedo, J.M.; Corona, A.S. Cryptanalysis of a Group Key Establishment Protocol. *Symmetry* **2021**, *13*, 332. https://doi.org/ 10.3390/sym13020332

Academic Editors: Juan Alberto Rodríguez Velázquez and Alejandro Estrada-Moreno

Received: 20 January 2021 Accepted: 9 February 2021 Published: 17 February 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Secure multiparty communication is an important concern for many current applications that work over public insecure channels, such as the Internet. Wireless sensor networks, collaborative applications, multiparty voice and video conferences, etc. need to guarantee confidentiality, integrity and authentication in their communications.

Group key establishment (GKE) protocols are fundamental in that sense. They allow a set of participants to agree on a common secret key to be used afterwards with symmetric key cryptographic primitives.

In some settings all the nodes play an equivalent role, and thus the group protocol is somewhat symmetric. Nevertheless, there are other applications where some nodes are distinguished and one can assume they may have more computational power and resources, and thus, they are required to perform more computations.

Over recent decades, group key establishment protocols were widely discussed in the literature [1–7], and formal security models were proposed, indicating which attacks the adversary can perform and what a secure key establishment protocol is. What is typically required is that, after completion of the protocol, the intended users agree on a common key, whereas the adversary does not learn anything about it.

A standard technique to augment the security of a scheme is the use of compilers, which allows a modular design, going from passively secure solutions to authenticated ones [8], from 2-party to group solutions [9], or adding forward secrecy [10].

However, several protocols were found to be insecure after they were published, because the proposals do not provide security proofs or the proofs are not correct [11–13]. Other protocols were found to be insecure when considering active attacks [14].

Motivated by the works in López-Ramos et al. [14], in this paper, we analyze a group key establishment proposal by López-Ramos et al. [15] and present several attacks on the proposed protocols. In particular, we present here some active attacks against the protocols, proving they are insecure when considering active adversaries.

*Contributions:* We present several concrete attacks showing the security flaws of the protocols proposed in López-Ramos et al. [15]. In Section 2, we review the proposal of López Ramos et al. Then, in Section 3 we review a standard security model for group key exchange. We then present our attacks in Section 4.

#### **2. The Protocol of López Ramos et al.**

In this section, we describe Protocol 1 in López-Ramos et al. [15], which can be seen as an extension of the classical 2-party Diffie-Hellman key exchange. Four different protocols are presented, which are modifications of this first one. In particular, Protocol 2 computes the same session key, but publishing only one public key and sending a different message in Round 2. Protocol 3 describe the extra steps to be done if some participants leave the group and Protocol 4 deals with the case where some users join the group.

#### Initialization

Let {*U*1, . . . , *Un*} be the finite set of protocol participants, including *Uc*<sup>1</sup> , who will act as controller. The users agree on a multiplicative cyclic group *G* of prime order *p* and on *g*, a generator of *G*.

Each user *U<sup>i</sup>* , 1 ≤ *i* ≤ *n* will have two random values, *r<sup>i</sup>* , *x<sup>i</sup>* ∈ Z ∗ *<sup>p</sup>* as private keys and *g <sup>r</sup><sup>i</sup>* and *g <sup>x</sup><sup>i</sup>* will be their public keys.

Round 1


#### Round 2

Every user *U<sup>i</sup>* , using the public information, computes *g* ∑*j*6=*i*,*c*<sup>1</sup> *<sup>r</sup><sup>j</sup>* and sends this value to *Uc*<sup>1</sup> (Notice that there is no need to send this information, since this value can be computed from the published public keys).

The group controller *Uc*<sup>1</sup> , moreover, computes

$$Y\_{1,i} = \mathcal{g}^{-\mathbf{x}\_{\mathbf{c}\_1}\mathbf{x}\_i} \left(\mathcal{g}^{r\_{\mathbf{c}\_1}\sum\_{j\neq c\_1,i}r\_j}\right) \quad \text{for} \quad i \in \{1,\ldots,n\} \ \{c\_1\} \quad \text{and}$$

$$Y\_{1,c\_1} = \mathcal{K}\_1 \mathcal{g}^{-r\_{c\_1}'r\_{c\_1}} \mathcal{g}^{-\mathbf{x}\_{c\_1}'\mathbf{x}\_{c\_1}},$$

$$\mathcal{R}\_1 = \mathcal{g}^{r\_{c\_1}} \quad \text{and} \quad \mathcal{S}\_1 = \mathcal{g}^{\mathbf{x}\_{c\_1}}.$$

He broadcasts (*Y*1,1, . . . ,*Y*1,*c*<sup>1</sup> , . . . ,*Y*1,*n*, *R*1, *S*1)

Key Computation

Once user *U<sup>i</sup>* has received the second round message, he computes the common session key *K*<sup>1</sup> := *K*1,*<sup>i</sup>* = *Y*1,*iS xi* 1 *R ri* 1 .

The protocol is summarized in Figure 1.

**Remark 1.** *The subindex* 1 *in the session key K*<sup>1</sup> *indicates here that it is the first execution of the protocol. In Protocols 3 and 4 in López-Ramos et al. [15], this subindex changes when the participants involved in the protocol change, i.e., some participants leave or join the protocol, and thus, some extra computations are needed.*

**Figure 1.** Protocol 1 of López Ramos et al.

#### **3. Security Model**

To formalize secure group key establishment, we use the somewhat standard Bohli et al.'s [5] security model, which builds on Jonathan Katz and Moti Yung [8].

#### *Security Goals: Semantic Security and Authentication*

#### Participants:

The (potential) protocol participants are modelled as probabilistic polynomial time (ppt) Turing machines in the finite set U = {*U*1, . . . , *Un*}. Each participant *U<sup>i</sup>* in the set U is able to run a polynomial amount of protocol instances in parallel.

We will refer to instance *s<sup>i</sup>* of principal *U<sup>i</sup>* as Π *si i* (*i* ∈ N) and it has the following variables assigned:


Communication network and adversarial capabilities:

We assume there exist arbitrary point to point connections among users and the network is non-private, fully asynchronous and in complete control of the adversary A, who can eavesdrop, delay, delete, modify or insert messages. The adversary's capabilities are captured by the following *oracles*:


We can distinguish two types of adversaries. An adversary with access to all the oracles described above is considered to be *active*. If the adversary is not granted access to any of the Send oracles, then it is considered a *passive* adversary.

To define semantic security, we also allow the adversary to have access to a Test oracle, which can be queried only once. The query Test(*U<sup>i</sup>* ,*si*) can be made on input an instance Π *si i* of user *U<sup>i</sup>* ∈ U only if acc *si <sup>i</sup>* = TRUE. In that case, a bit *b* ← {0, 1} is chosen uniformly at random; if *b* = 0, the oracle returns the session key stored in sk*s<sup>i</sup> i* . Otherwise, the oracle outputs a uniformly at random chosen element from the space of session keys.

#### Security notions:

For the schemes to be useful, we need the group key establishments to be *correct*, i.e., without adversarial interference, the protocol would allow all users to compute the same key.

**Definition 1** (correctness)**.** *A group key establishment is* correct *if for all instances* Π *si i ,* Π *sj j which accepted with* sid*s<sup>i</sup> <sup>i</sup>* <sup>=</sup> sid*s<sup>j</sup> j and* pid*s<sup>i</sup> <sup>i</sup>* <sup>=</sup> pid*s<sup>j</sup> j , the condition* sk*s<sup>i</sup> <sup>i</sup>* <sup>=</sup> sk*s<sup>j</sup> j* 6= NULL *is satisfied.*

To be more precise in the security definition, it is important to specify under which conditions the adversary can query the Test oracle. To do so, we first define the following notion of *partnering*:

**Definition 2** (partnering)**.** *Two terminated instances* Π *si i and* Π *sj j are* partnered *if* sid*s<sup>i</sup> <sup>i</sup>* <sup>=</sup> sid*s<sup>j</sup> j ,* pid*s<sup>i</sup> <sup>i</sup>* <sup>=</sup> pid*s<sup>j</sup> j and* acc *si Ui* = acc *sj Uj* = TRUE*.*

To avoid queries that would trivially allow the adversary to know the key, we restrict the instances that can be queried to the Test oracle, only allowing *fresh* instances:

**Definition 3** (freshness)**.** *We say an instance* Π *si i is* fresh *if none of the following events has occurred:*


**Remark 2.** *The previous definition for freshness allows including the desired goal of* forward secrecy *in our definition of security given below: an adversary* A *is allowed to query* Corrupt *for* all *users and obtain their long term keys without violating freshness, if he does not send any message afterwards.*

Let Succ<sup>A</sup> be the event that the adversary A queries the Test oracle with a fresh instance and makes a correct guess about the random bit *b* used by the Test oracle, we define the *advantage* of an adversary A attacking protocol *P* as

$$\text{Adv}\_{\mathcal{A}}^{\text{ke}} = \text{Adv}\_{\mathcal{A}}^{\text{ke}}(k) := \left| \Pr[\mathsf{Succ}\_{\mathcal{A}}] - \frac{1}{2} \right|.$$

**Definition 4** (semantic security)**.** *A group key establishment protocol is* (semantically) secure*, if* Advke <sup>A</sup> <sup>=</sup> Advke <sup>A</sup> (*k*) *is negligible for every ppt adversary* <sup>A</sup>*.*

#### **4. Cryptanalysis of the Proposal of López-Ramos et al.**

In this section, we describe several concrete attacks refuting the security results of López-Ramos et al. [15], where four different, but related, GKE protocols are described. The four protocols will be considered in this section. However, we will only explicitly attack Protocol 1, being the attacks to the others straightforwardly adapted.

#### *4.1. Active Attack*

Informally, since the protocol is not authenticated, we will describe here how an adversary can attack the protocol by mounting a Man-In-The-Middle attack. Users will end up sharing a key with the adversary, instead of with all the intended communication partners. We formalize the attack below.

Let us fix {*U*1, ..., *Un*} the set of communication parties and let A be an active attacker able to supersede some parties in the set. We will distinguish two different cases: A shares a key with the group controller *Uc*<sup>1</sup> and other with the rest of the users and A shares a key with any other party *U<sup>i</sup>* , *i* 6= *c*1, and a different key with the rest, including the controller.

If A tries to share a different key with the group controller *Uc*<sup>1</sup> the adversary can build an attack by following the next steps:


Notice that every user *U<sup>i</sup>* , *i* 6= 1, *c*1, after receiving that message, will compute and *n* ∑ *rj*

send the value *g j*=1,*j*6=*c*1 and therefore this value will be output by the Send oracle.

The controller *Uc*<sup>1</sup> , after receiving that message, will compute and send the value *a*1+ *n* ∑ *rj*


$$Z\_{1,i} = \mathcal{g}^{-b\_{c\_1}x\_i} \mathcal{g}^{a\_{c\_1} \left(\sum\_{j=1, j\neq c\_1, i}^n r\_j\right)} \quad i \neq c\_1$$

$$Z\_{1,c\_1} = Q\_1 \mathcal{g}^{-a\_{c\_1}' a\_{c\_1}} \mathcal{g}^{-b\_{c\_1}' b\_{c\_1}}.$$

*n*

5. The adversary A will query Send(*U<sup>i</sup>* ,*si* ,(*Z*1,1, . . . , *Z*1,*n*, *T*1, *V*1) oracle, for all *i* ∈ {1, ..., *n*} \{*c*1}.

6. The adversary A will compute the session key *K*<sup>1</sup> = *g rc*1 ( ∑ *j*=1,*j*6=*c*1 *rj* ) and the values *R*<sup>1</sup> = *g <sup>r</sup>c*<sup>1</sup> and *S*<sup>1</sup> = *g <sup>x</sup>c*<sup>1</sup> , along with the keying values

*n*

)

$$Y\_{1,i} = \mathcal{g}^{-\mathbf{x}\_{c\_1}\mathbf{x}\_i} \mathcal{g}^{r\_{c\_1}(a\_1 + \sum\_{j=2, j\neq c\_1, i}^n r\_j)} \quad i \neq c\_1.$$

$$Y\_{1,c\_1} = \mathcal{K}\_1 \mathcal{g}^{-r'\_{c\_1}r\_{c\_1}} \mathcal{g}^{-\mathbf{x}'\_{c\_1}\mathbf{x}\_{c\_1}}.$$

7. The adversary A will query Send(*U*1,*s*1,(*Y*1,1, . . . ,*Y*1,*n*, *S*1, *T*1) oracle.

Please note that after receiving this last message, users {*U*1, . . . , *Un*} \ {*Uc*<sup>1</sup> }, following the protocol, will compute *Q*1,*<sup>i</sup>* = *Z*1,*iT xi* <sup>1</sup> *V ri* 1 . Please note that *Q*1,*<sup>i</sup>* = *Q*<sup>1</sup> for every *i* 6= *c*1.

On the other hand, the group controller *Uc*<sup>1</sup> will compute *K*<sup>1</sup> = *Y*1,1*S b*1 1 *R a*1 1 .

Therefore, after this attack, the adversary has established a shared key *Q*<sup>1</sup> with the set of parties {*U*1, . . . , *Un*} \ {*Uc*<sup>1</sup> } and the key *K*<sup>1</sup> with the group controller *Uc*<sup>1</sup> , where

$$Q\_1 = \stackrel{a\_{\mathcal{L}\_1}}{\text{g}}^{\text{a}\_{\mathcal{L}\_1}} \stackrel{\sum^n}{\text{ }}^{r\_j} \quad \text{and} \quad K\_1 = \stackrel{r\_{\mathcal{L}\_1}(a\_1 + \sum^n\_{j = 2, j \neq \mathcal{c}\_1} r\_j)}{\text{ }} \dots$$

Consequently, all the users will believe they are establishing a common key when they are not. Moreover, the adversary can decrypt the messages sent encrypted with both keys and forward the communication between the users that do not share a key.

This attack is outlined in Figure 2.

If A tries to compute a different key with any user different from the group controller, we can assume without loss of generality that A is sharing it with *U*1. The adversary A can build an attack following the subsequent steps:


Notice that every user *U<sup>i</sup>* , *i* 6= 1, *c*<sup>1</sup> and the adversary A, after receiving that message, (*a*1+ *n* ∑ *rj* )

will compute *g j*=2,*j*6=*c*1 and therefore this value will be output by the Send oracle. *n*

Moreover, the group controller *Uc*<sup>1</sup> will calculate the session key *Q*<sup>1</sup> = *g rc*1 (*a*1+ ∑ *j*=2,*j*6=*c*1 *rj* and he will send *R*<sup>1</sup> = *g <sup>r</sup>c*<sup>1</sup> and *S*<sup>1</sup> = *g <sup>x</sup>c*<sup>1</sup> , along with the keying values

$$Z\_{1,1} = \mathcal{g}^{-\mathbf{x}\_{c\_1} b\_1} \mathcal{g}^{(r\_{c\_1} \sum\_{j=2, j \neq c\_1}^n r\_j)}.$$

$$Z\_{1,i} = \mathcal{g}^{-\mathbf{x}\_{c\_1} \mathbf{x}\_i} \mathcal{g}^{r\_{c\_1}(a\_1 + \sum\_{j=2, j \neq c\_1}^n r\_j)} \quad i \neq 1, c\_1.$$

$$Z\_{1, \mathcal{L}\_1} = Q\_1 \mathcal{g}^{-r\_{c\_1}' r\_{c\_1}} \mathcal{g}^{-\mathbf{x}\_{c\_1}' \mathbf{x}\_{c\_1}}.$$

These values will also be part of the output of the Send oracle.

Please note that after receiving this message every user *U<sup>i</sup>* , *i* 6= 1, can compute the key *Q*<sup>1</sup> = *Z*1,*iS xi* 1 *R ri* 1 that will be shared with the adversary A.

**Figure 2.** Active attack on Protocol 1 of López Ramos et al.

4. The attacker A will delete the message sent by *Uc*<sup>1</sup> to the superseded user *U*1, and queries Send(*U*1,*s*1,(*W*1,1, . . . , *W*1,*n*, *T*1, *V*1)), where

$$W\_{1,i} = \mathcal{g}^{-b\_{c\_1}x\_i} \mathcal{g}^{a\_{c\_1}(\sum\_{j=1, j\neq i, c\_1}^n r\_j)},$$

$$W\_{1,c\_1} = K\_1 \mathcal{g}^{-a'\_{c\_1}a\_{c\_1}} \mathcal{g}^{-b'\_{c\_1}b\_{c\_1}},$$

$$T\_1 = \mathcal{g}^{a\_{c\_1}} \quad \text{and} \quad V\_1 = \mathcal{g}^{b\_{c\_1}}.$$

Please note that user *U*1, after receiving these last messages, can compute the key *K*<sup>1</sup> = *W*1,1*T x*1 <sup>1</sup> *V r*1 <sup>1</sup> which is shared with the adversary A.

	- (a) The superseded user *U*<sup>1</sup> will compute *K*<sup>1</sup> = *W*1,1*T x*1 <sup>1</sup> *V r*1 1 .
	- (b) Every user *U<sup>i</sup>* , *i* 6= 1 computes *Q*1,*<sup>i</sup>* = *Y*1,*iS xi* 1 *R ri* 1 .
	- (c) Adversary A computes *Q*1,1 = *Z*1,1*S b*1 1 *R a*1 1 and *K*<sup>1</sup> = *W*1,1*T x*1 <sup>1</sup> *V r*1 1 .

Therefore, the adversary A has established a shared key *Q*<sup>1</sup> with the set of parties {*U*2, ..., *Un*}. On the other hand, both *U*<sup>1</sup> and the adversary A share the common key *K*1.

**Remark 3.** *While in López-Ramos et al. [15] four different protocols were described, in the previous lines only Protocol 1 was attacked.*

*In Protocol 2, authors try to share the computational requirements in a more even way among the parties by slightly modifying which values every participant sends to the group controller and the computations that this user has to perform. However, the only private information for every user is the tuple* (*r<sup>i</sup>* , *xi*) *as in Protocol 1. Thus, an attack can be built analogously by following the steps described above.*

*In Protocol 3, authors assume that the group controller has changed. The new group controller, by using two private elements* (*r* 0 *ct* , *x* 0 *ct* ) *makes a transformation of the key. The next steps of Protocol 3 follows the description of Protocol 1. Therefore, an attack can be built following the previous description.*

*In Protocol 4, new users take part in the round with new private elements* (*r<sup>t</sup>* , *xt*)*. Therefore a new key has to be computed by the group controller using those new elements. Once more, subsequent steps of Protocol 4 follows the description of Protocol 1 and an attack can be constructed analogously.*

#### *4.2. Forward Secrecy*

We will informally describe how a passive adversary who corrupts a participant *U<sup>i</sup>* ∈ {*U*1, . . . , *Un*} involved in a protocol run will be able to compute the shared session key. Therefore, the protocol does not provide forward secrecy.

Let A be a probabilistic polynomial time adversary (modelled as a Turing machine). He may perform an attack by following the next steps:


Please note that session *s<sup>j</sup>* of user *U<sup>j</sup>* remains fresh, since, the adversary has not made any Send or Reveal query, so the attack is legitimate.

**Remark 4.** *In Protocols 3 and 4 in López-Ramos et al. [15], it is described how to proceed when participants may join or leave the group. However, when a participant leaves, the only user changing his private and public keys is the new controller. This means that the rest of the users will have the same private and public key used for previous instances. Therefore, when corrupting any user that is not the new controller, one will obtain their private keys and mount the attack described above. Protocol 2, can also be attacked in the same way, just changing the computations to obtain the session key according to the protocol description.*

**Remark 5.** *As observed in Theorem 2.4 in López-Ramos et al. [15], the keying messages sent to establish the key can be seen as ElGamal-like encryptions of the key K*<sup>1</sup> *under a different key for* *each user. In that sense, the protocol can be interpreted as a key transport protocol, which cannot be forward secret.*

**Remark 6.** *Countermeasures: If a security proof of the protocols in López-Ramos et al. [15] is provided for passive adversaries, and one consider the private keys as random nonces to be used only in one instance of the protocol, one could then apply the compiler in Katz and Yung [8] to avoid active attacks, generating long term keys for each user to compute digital signatures on all the exchanged messages to guarantee authentication. In that case, if the keys are nonces,* Corrupt *oracle queries would return the signing private keys, thus, corrupted users would not be able to compute the session keys and forward secrecy would also be granted.*

#### **5. Conclusions**

As demonstrated above, the protocol proposed by López Ramos et al. [15] does not offer security guarantees. The paper does not provide a rigorous security proof in any standard security model using provable security techniques. The proofs provided are too schematic. If a compiler for authentication is used and the private keys are ephemeral, some attacks could not be applicable. Nevertheless, a security proof should be provided.

**Author Contributions:** Individual contributions to this article: conceptualization, J.M.C. and A.S.C.; methodology, J.M.C. and A.S.C.; validation, J.M.C. and A.S.C.; formal analysis, J.M.C. and A.S.C.; software, J.M.C. and A.S.C.; investigation, J.M.C. and A.S.C.; resources, J.M.C. and A.S.C.; writing original draft preparation, J.M.C. and A.S.C.; writing–review and editing, J.M.C. and A.S.C.; project administration, J.M.C. and A.S.C.; funding acquisition, J.M.C. and A.S.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded in part through research project MTM2017-83506-C2-2-P by the Spanish MICINN.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs**

**Ana Almerich-Chulia 1,† , Abel Cabrera Martínez 2,\* ,† , Frank Angel Hernández Mira 3,† and Pedro Martin-Concepcion 1,†**


**Abstract:** Let *G* be a graph with no isolated vertex and let *N*(*v*) be the open neighbourhood of *v* ∈ *V*(*G*). Let *f* : *V*(*G*) → {0, 1, 2} be a function and *V<sup>i</sup>* = {*v* ∈ *V*(*G*) : *f*(*v*) = *i*} for every *i* ∈ {0, 1, 2}. We say that *f* is a strongly total Roman dominating function on *G* if the subgraph induced by *V*<sup>1</sup> ∪ *V*<sup>2</sup> has no isolated vertex and *N*(*v*) ∩ *V*<sup>2</sup> 6= ∅ for every *v* ∈ *V*(*G*) \ *V*2. The strongly total Roman domination number of *G*, denoted by *γ s tR*(*G*), is defined as the minimum weight *<sup>ω</sup>*(*f*) = <sup>∑</sup>*x*∈*V*(*G*) *f*(*x*) among all strongly total Roman dominating functions *f* on *G*. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing *γ s tR*(*G*) is NP-hard.

**Keywords:** strongly total Roman domination; total Roman domination; total domination; lexicographic product graph

## **1. Introduction**

Let *G* be a simple graph with no isolated vertex. Given a vertex *v* ∈ *V*(*G*), *N*(*v*) and *N*[*v*] denote the *open neighbourhood* and the *closed neighbourhood* of *v* in *G*, respectively. The *order*, *minimum degree* and *maximum degree* of *G* will be denoted by *n*(*G*), *δ*(*G*) and ∆(*G*), respectively. As usual, given a set *D* ⊆ *V*(*G*) and a vertex *v* ∈ *D*, the *external private neighbourhood* and the *internal private neighbourhood* of *v* with respect to *D* is defined to be *epn*(*v*, *D*) = {*u* ∈ *V*(*G*) \ *D* : *N*(*u*) ∩ *D* = {*v*}} and *ipn*(*v*, *D*) = {*u* ∈ *D* : *N*(*u*) ∩ *D* = {*v*}}, respectively.

Domination in graphs is a classical research topic that has experienced rapid growth since its introduction. A set *D* ⊆ *V*(*G*) is said to be a *dominating set* of *G* if *N*(*v*) ∩ *D* 6= ∅ for every *v* ∈ *V*(*G*) \ *D*. Let D(*G*) be the set of dominating sets of *G*. The *domination number* of *G* is defined to be the following.

$$\gamma(G) = \min\{|D| \colon D \in \mathcal{D}(G)\}.$$

We define a *γ*(*G*)-set as a set *D* ∈ D(*G*) with |*D*| = *γ*(*G*). The same agreement will be assumed for optimal parameters associated with other characteristic functions or sets of

**Citation:** Almerich-Chulia, A.; Cabrera Martínez, A.; Hernández Mira, F.A.; Martin-Concepcion, P. From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs. *Symmetry* **2021**, *13*, 1282. https://doi.org/10.3390/sym13071282

Academic Editors: Markus Meringer and Michel Planat

Received: 17 June 2021 Accepted: 15 July 2021 Published: 16 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

a graph. For more information on domination and its variants in graphs, we suggest the books [1–4].

An important domination variant in graph, which may be the most studied, is the total domination number. A *total dominating set* of *G* is a set *D* ∈ D(*G*) such that *N*(*v*) ∩ *D* 6= ∅ for every *v* ∈ *D*. Let D*t*(*G*) be the set of total dominating sets of *G*. The *total domination number* of *G* is defined to be the following.

$$\gamma\_t(G) = \min\{|D| \, : \, D \in \mathcal{D}\_t(G)\}.$$

More information on total domination in graphs can be found in the works [5–7]. Let *G* be a graph with no isolated vertex and *f* : *V*(*G*) → {0, 1, 2} a function. For every *i* ∈ {0, 1, 2}, we define *V<sup>i</sup>* = {*v* ∈ *V*(*G*) : *f*(*v*) = *i*}. We will use the unified notation *f*(*V*0, *V*1, *V*2) to identify the function *f* with the subsets *V*0, *V*1, *V*<sup>2</sup> associated with it. Given a set *<sup>X</sup>* ⊆ *<sup>V</sup>*(*G*), we define *<sup>f</sup>*(*X*) = <sup>∑</sup>*x*∈*<sup>X</sup> <sup>f</sup>*(*x*) and, particularly, we define the *weight* of *<sup>f</sup>* as *ω*(*f*) = *f*(*V*(*G*)) = |*V*1| + 2|*V*2|. One of the domination topics widely studied by research is the Roman domination, which is a domination variant arising from some historical roots coming from the ancient Roman Empire [8]. A function *f*(*V*0, *V*1, *V*2) is a *Roman dominating function* on *G* if *N*(*v*) ∩ *V*<sup>2</sup> 6= ∅ for every vertex *v* ∈ *V*0. The *Roman domination number* of *G* denoted by *γR*(*G*) is the minimum weight among all Roman dominating functions on *G*. For more information on Roman domination in graphs, we suggest the referenced works [9–12].

One of the classical variants of Roman domination is the so-called total Roman domination. This article deals precisely with this style of domination. A *total Roman dominating function* (TRDF) on a graph *G* with no isolated vertex is a Roman dominating function *f*(*V*0, *V*1, *V*2) such that *V*<sup>1</sup> ∪ *V*<sup>2</sup> ∈ D*t*(*G*). The minimum weight among all TRDFs on *G* is the *total Roman domination number* of *G* and is denoted as *γtR*(*G*). This concept was introduced in 2013 by Liu and Chang [13] and formally presented and deeply studied three years later by Abdollahzadeh Ahangar et al. [14]. Subsequently, several researchers have continued with the study of this parameter. For instance, in [15–17], some combinatorial results were presented. In [18–21], constructive characterizations in trees related with this domination parameter were provided. In [22–25], studies of the total Roman domination number on graph products were carried out. In particular, we want to highlight the following closed formula provided in [25] for the case of lexicographic product graphs.

For any graph *G* with no isolated vertex and any nontrivial graph *H*, the total Roman domination number of the lexicographic product graph *G* ◦ *H* is given by the following [25]:

$$\gamma\_{t\mathbb{R}}(G\circ H) = \begin{cases} \ 2\gamma\_t(G) & \text{if } \gamma(H) \ge 2, \\\ \int \zeta(G) & \text{if } \gamma(H) = 1, \end{cases} \tag{1}$$

where *ξ*(*G*) = min{|*A*| + 2|*B*| : *B* ∈ D(*G*) and *A* ∪ *B* ∈ D*t*(*G*)}. As it can be observed, the authors [25] showed that the behavior of *γtR*(*G* ◦ *H*) depends on two domination parameters of graphs, namely the well-known total domination number and the incipient parameter *ξ*(*G*). In that regard, the authors exposed some results on this last parameter and they raised the challenge of conducting a detailed study of the topic.

In this paper, we continue with the study of this novel parameter although it will be carried out by considering a different approach. In Section 2 we define a new variant of total Roman domination, namely strongly total Roman domination number and denoted by *γ s tR*(*G*). We then show that this variant is an appropriate framework to investigate the parameter *ξ*(*G*) of a graph. Section 3 is devoted to providing closed formulas for some product graphs. As a consequence of the study, we conclude this section by showing that the problem of computing *γ s tR*(*G*) is NP-hard. Finally, in Section 4 we obtain tight bounds on the strongly total Roman domination number of a graph and we discuss the tightness of these bounds.

We assume that the reader is familiar with the basic concepts and terminology of graph domination. If this is not the case, then we suggest the textbooks [1,4]. For the remainder of the article, definitions will be introduced whenever a concept is required.

#### **2. Strongly Total Roman Dominating Functions**

The concept of a total Roman dominating function on a graph is associated with the "total domination" property, i.e., this kind of functions requires that each vertex has a neighboring vertex with a positive label assigned to it. However, some vertices have a "special property", which in some cases others do not have. In particular, vertices with label zero must always have a neighbor with label two, but it is not always the case that a vertex with label one satisfies this property. In relation to the above situation, we introduce a "stronger" version of the standard total Roman domination below.

A *strongly total Roman dominating function* (STRDF) on a graph *G* with no isolated vertex is a total Roman dominating function *f*(*V*0, *V*1, *V*2) with the additional property that *V*<sup>2</sup> is a dominating set of *G*. The minimum weight among all STRDFs on *G* is the *strongly total Roman domination number* of *G* and is denoted *γ s tR*(*G*).

To illustrate this concept, we consider the graph *G* shown in Figure 1. For this example, *γtR*(*G*) < *γ s tR*(*G*).

**Figure 1.** The function on the left is a *γtR*(*G*)-function, while the function on the right is a *γ s tR*(*G*)-function.

Now, we proceed to show that this new domination variant is an appropriate framework to investigate the parameter *ξ*(*G*).

**Theorem 1.** *For any graph G with no isolated vertex,*

$$
\gamma\_{tR}^s(G) = \mathfrak{f}(G).
$$

**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γ s tR*(*G*)-function. By definition we have that *V*<sup>2</sup> ∈ D(*G*) and *V*<sup>1</sup> ∪ *V*<sup>2</sup> ∈ D*t*(*G*). Therefore, the following obtains.

$$\mathcal{L}(G) = \min\{|A| + 2|B| \, : \, B \in \mathcal{D}(G) \text{ and } A \cup B \in \mathcal{D}\_l(G)\} \le |V\_1| + 2|V\_2| = \gamma\_{t\mathcal{R}}^s(G).$$

On the other side, let *A* 0 , *B* <sup>0</sup> ⊆ *V*(*G*) such that *B* <sup>0</sup> ∈ D(*G*), *A* <sup>0</sup> ∪ *B* <sup>0</sup> ∈ D*t*(*G*) and *ξ*(*G*) = |*A* 0 | + 2|*B* 0 |. Notice that the function *f* 0 (*V* 0 0 , *V* 0 1 , *V* 0 2 ), defined by *V* 0 <sup>2</sup> = *B* 0 , *V* 0 <sup>1</sup> = *A* 0 and *V* 0 <sup>0</sup> = *V*(*G*) \ (*A* <sup>0</sup> ∪ *B* 0 ), is a STRDF on *G*. Hence, *γ s tR*(*G*) ≤ *ω*(*f* 0 ) = |*A* 0 | + 2|*B* 0 | = *ξ*(*G*), which completes the proof.

To end this section and as a consequence of previous theorem, we show the basic results given in [25] for the strongly total Roman domination number.

**Theorem 2.** Ref. [25] *For any graph G with no isolated vertex,*

$$
\max\{\gamma\_{tR}(G), \gamma\_t(G) + \gamma(G)\} \le \gamma\_{tR}^s(G) \le \min\{3\gamma(G), 2\gamma\_t(G)\}.
$$

*Furthermore,*


#### **3. Exact Formulas for Some Graph Products and Computational Complexity**

In order to show the tightness of several bounds and relationships, in this section we obtain the strongly total Roman domination number concerning a well-know families of graphs. We emphasize that we will use the notation *Kn*, *K*1,*n*−1, *Kr*,*n*−*<sup>r</sup>* and *W<sup>n</sup>* for complete graphs, star graphs, complete bipartite graphs and the wheel graphs of order *n*, respectively.

The join graph *G* + *H* of the graphs *G* and *H* is the graph with vertex set *V*(*G* + *H*) = *V*(*G*) ∪ *V*(*H*) and edge set *E*(*G* + *H*) = *E*(*G*) ∪ *E*(*H*) ∪ {*uv* : *u* ∈ *V*(*G*), *v* ∈ *V*(*H*)}.

**Theorem 3.** *For any graphs G and H,*

$$\gamma\_{tR}^s(G+H) = \begin{cases} 3 & \text{if } \gamma(G) = 1 \text{ or } \gamma(H) = 1, \\\ 4 & \text{otherwise.} \end{cases}$$

**Proof.** We first notice that *γt*(*G* + *H*) = 2. Now, we observe that *γ*(*G* + *H*) = 1 if and only if *γ*(*G*) = 1 or *γ*(*H*) = 1. Therefore, by Theorem 2 we deduce that *γ s tR*(*G* + *H*) = 3 if and only if *γ*(*G*) = 1 or *γ*(*H*) = 1, which completes the proof.

The following corollary is an immediate consequence of the theorem above.

**Corollary 1.** *The following equalities hold for any integer n* ≥ 3*.*


Let *G* be a graph with no isolated vertex and *H* is any graph. The corona product graph *G H* is defined as the graph obtained from *G* and *H*, by taking one copy of *G* and |*V*(*G*)| copies of *H* and joining by an edge every vertex from the *i*th-copy of *H* with the *i*th-vertex of *G*. Next, we study the strongly total Roman domination number of any corona product graph.

**Theorem 4.** *For any graph G with no isolated vertex and any graph H,*

$$
\gamma\_{tR}^s(G \odot H) = 2n(G).
$$

**Proof.** First, we notice that *γt*(*G H*) = *γ*(*G H*) = *n*(*G*). Hence, Theorem 2 leads to the equality *γ s tR*(*G H*) = 2*n*(*G*). Therefore, the proof is complete.

Let *G* be a graph with no isolated vertex and *H* a nontrivial graph. The *lexicographic product* of *G* and *H* is the graph *G* ◦ *H* for which the vertex set is *V*(*G* ◦ *H*) = *V*(*G*) × *V*(*H*) and two vertices (*u*, *v*),(*x*, *y*) ∈ *V*(*G* ◦ *H*) are neighbors if and only if *ux* ∈ *E*(*G*) or *u* = *x* and *vy* ∈ *E*(*H*).

**Theorem 5.** Ref. [26] *For any graph G with no isolated vertex and any nontrivial graph H,*

$$
\gamma\_t(G \circ H) = \gamma\_t(G).
$$

We next show that the strongly total Roman domination number and the total Roman domination number coincide for lexicographic product graphs.

**Theorem 6.** *For any graph G with no isolated vertex and any nontrivial graph H,*

$$\gamma\_{tR}^s(G\circ H) = \begin{cases} 2\gamma\_t(G) & \text{if } \gamma(H) \ge 2. \\\ \gamma\_{tR}^s(G) & \text{if } \gamma(H) = 1. \end{cases}$$

**Proof.** If *γ*(*H*) ≥ 2, then the result immediately follows by applying Equation (1) and Theorems 2 and 5, i.e., we have the following.

$$2\gamma\_t(G) = \gamma\_{tR}(G \circ H) \le \gamma\_{tR}^s(G \circ H) \le 2\gamma\_t(G \circ H) = 2\gamma\_t(G).$$

From this moment on, we assume that *γ*(*H*) = 1. By Equation (1) and Theorems 1 and 2 we deduce that *γ s tR*(*G*) = *ξ*(*G*) = *γtR*(*G* ◦ *H*) ≤ *γ s tR*(*G* ◦ *H*). We only need to prove that *γ s tR*(*G* ◦ *H*) ≤ *γ s tR*(*G*). Let *f*(*V*0, *V*1, *V*2) be a *γ s tR*(*G*)-function and {*v*} a *γ*(*H*)-set. Notice that the function *g*(*W*0, *W*1, *W*2), defined by *W*<sup>2</sup> = *V*<sup>2</sup> × {*v*}, *W*<sup>1</sup> = *V*<sup>1</sup> × {*v*} and *W*<sup>0</sup> = *V*(*G* ◦ *H*) \ (*W*<sup>1</sup> ∪ *W*2), is a STRDF on *G* ◦ *H*. Hence, *γ s tR*(*G* ◦ *H*) ≤ |*W*1| + 2|*W*2| = |*V*1| + 2|*V*2| = *γ s tR*(*G*), which completes the proof.

As shown in [27], the general optimization problem of computing the total domination number of a graph with no isolated vertex is NP-hard. Therefore, by Theorem 6 (considering the case *γ*(*H*) ≥ 2) we immediately obtain the analogous result for the strongly total Roman domination number.

**Theorem 7.** *The problem of computing the strongly total Roman domination number of a graph with no isolated vertex is NP-hard.*

#### **4. Primary Combinatorial Results**

The first result of this section provides bounds for the strongly total Roman domination number in terms of the order of a graph. For this purpose, we need to recall the following well-known result.

**Theorem 8.** Ref. [5] *If G is a connected non-complete graph of order at least three, then G has a γt*(*G*)*-set D such that every vertex v* ∈ *D satisfies epn*(*v*, *D*) 6= ∅ *or is adjacent to a vertex v* <sup>0</sup> ∈ *ipn*(*v*, *D*) *satisfying epn*(*v* 0 , *D*) 6= ∅*.*

**Theorem 9.** *For any connected graph G of order at least three,*

$$\mathfrak{Z} \le \gamma\_{tR}^s(G) \le n(G).$$

*Furthermore,*

$$\text{(i)}\qquad \gamma\_{tR}^s(G) = \mathfrak{Z}\text{ if and only if }\gamma(G) = 1.$$

(ii) *γ s tR*(*G*) = 4 *if and only if γt*(*G*) = *γ*(*G*) = 2*.*

**Proof.** The lower bound is straightforward. Now, we proceed to prove the upper bound. If *G* is isomorphic to a complete graph, then *γ s tR*(*G*) = 3 ≤ *n*(*G*), as desired. From this moment, we assume that *G* is different of a complete graph. Let *D* be a *γt*(*G*)-set which satisfies Theorem 8 and *D* = *V*(*G*) \ *D*. Now, we consider the following sets.

$$D\_{\varepsilon} = \{ v \in D : \operatorname{epn}(v, D) \neq \mathcal{Q} \} \quad \text{and} \quad \overline{D\_{\varepsilon}} = \{ v \in \overline{D} : N(v) \cap D\_{\varepsilon} \neq \mathcal{Q} \}.$$

Let us define *f* 0 (*V* 0 0 , *V* 0 1 , *V* 0 2 ) as a function of minimum weight among all functions *f*(*V*0, *V*1, *V*2) on *G* satisfying the following conditions.


By (a), it is straightforward that *V* 0 <sup>1</sup> ∪ *V* 0 <sup>2</sup> ∈ D*t*(*G*). By (b) and (c) we deduce that every vertex in *V* 0 <sup>0</sup> = *D* has a neighbor in *V* 0 2 . Now, let *v* ∈ *V* 0 1 . By definition, *v* ∈ *D* \ *D<sup>e</sup>* and thus Theorem 8 results in *N*(*v*) ∩ *D<sup>e</sup>* 6= ∅. Hence, *N*(*v*) ∩ *V* 0 2 6= ∅ by (b). This implies that *V* 0 <sup>2</sup> ∈ D(*G*). Therefore, *f* 0 is a STRDF on *G* and thus *γ s tR*(*G*) ≤ *ω*(*f* 0 ).

We only need to prove that *ω*(*f* 0 ) ≤ *n*(*G*). Let *v* ∈ *D* \ *D<sup>e</sup>* . By definition, we have that *N*(*v*) ∩ *D* ⊆ *D* \ *D<sup>e</sup>* and |*N*(*v*) ∩ *D*| ≥ 2. Hence, by (a) and (c) we deduce that *N*(*v*) ∩ *V* 0 2 \ *D<sup>e</sup>* 6= ∅. Thus, the minimality of *f* 0 results in |*V* 0 2 \ *D<sup>e</sup>* | ≤ |*D* \ *D<sup>e</sup>* | and it is straightforward that |*V* 0 <sup>2</sup> ∩ *D<sup>e</sup>* | ≤ |*D<sup>e</sup>* | ≤ |*D<sup>e</sup>* |. Therefore, the following

$$\begin{aligned} \omega(f') &= |V\_1'| + 2|V\_2'|\\ &= |D| + |V\_2' \backslash D\_\varepsilon| + |V\_2' \cap D\_\varepsilon| \\ &\le |D| + |\overline{D} \backslash \overline{D\_\varepsilon}| + |\overline{D\_\varepsilon}| \\ &= |D| + |\overline{D}| \\ &= n(G)\_\prime \end{aligned}$$

is as required. Hence, the proof of the upper bound is complete.

We then proceed to prove (i). By Theorem 2 we deduce that *γ s tR*(*G*) = 3 if and only if *γ*(*G*) = 1. Hence, (i) follows.

Finally, we proceed to prove (ii). If *γt*(*G*) = *γ*(*G*) = 2, then Theorem 2 leads to *γ s tR*(*G*) = 4. Conversely, if *γ s tR*(*G*) = 4, then by (i) we deduce that *γ*(*G*) ≥ 2. Thus, Theorem 2 results in *γt*(*G*) = *γ*(*G*) = 2. Therefore, (ii) follows.

The upper bound above is tight. For instance, it is achieved for the graph *G* given in Figure 1. Moreover and as an immediate consequence of Theorems 2 and 9, it is also achieved for the graphs *G* with *γtR*(*G*) = *n*(*G*). This family is defined in [14].

We continue by providing additional upper bounds. As shown in Theorem 2, the strongly total Roman domination number is bounded from above by 3*γ*(*G*). Since *γR*(*G*) ≤ 2*γ*(*G*), the next result improves this upper bound for any graph *G* with no isolated vertex. We need to introduce the following result.

**Theorem 10.** Ref. [9] *Let f*(*V*0, *V*1, *V*2) *be a γR*(*G*)*-function on a graph G with no isolated vertex such that* |*V*1| *is minimum. Then the following conditions hold.*

(a) *N*(*v*) ⊆ *V*<sup>0</sup> *for every vertex v* ∈ *V*1*.*

(b) *N*[*x*] ∩ *N*[*y*] = ∅ *for any two different vertices x*, *y* ∈ *V*1*.*

**Theorem 11.** *For any graph G with no isolated vertex,*

$$
\gamma\_{tR}^s(G) \le \gamma\_R(G) + \gamma(G).
$$

**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γR*(*G*)-function such that |*V*1| is minimum. Hence, conditions (a) and (b) of Theorem 10 are satisfied. Now, we consider a function *g* 0 (*W*0 0 , *W*0 1 , *W*0 2 ) of minimum weight among all functions *g*(*W*0, *W*1, *W*2) on *G* such that the following conditions are satisfied:


We proceed to prove that *g* 0 is a STRDF on *G*. By definitions of *f* and *g* 0 , it is straightforward that *W*0 <sup>1</sup> ∪ *W*<sup>0</sup> <sup>2</sup> ∈ D*t*(*G*). Now, let *v* ∈ *V*(*G*) \ *W*<sup>0</sup> 2 . By (i) we deduce that *v* ∈ *V*<sup>0</sup> ∪ *V*1. Moreover, if *v* ∈ *V*0, then *N*(*v*) ∩ *W*<sup>0</sup> 2 6= ∅ because *N*(*v*) ∩ *V*<sup>2</sup> 6= ∅. Otherwise, if *v* ∈ *V*1, then (ii) results in *N*(*v*) ∩ *W*<sup>0</sup> 2 6= ∅. Hence, *W*<sup>0</sup> <sup>2</sup> ∈ D(*G*), which implies that *g* 0 is a STRDF on *G*, as desired. Thus, *γ s tR*(*G*) ≤ *ω*(*g* 0 ).

Now, let *D* be a *γ*(*G*)-set. Hence, *N*[*v*] ∩ *D* 6= ∅ for every *v* ∈ *V*(*G*). In addition, notice that *N*(*v*) ∩ (*W*<sup>0</sup> 2 \ *V*2) 6= ∅ for every *v* ∈ *V*1. Thus, by (ii) and (iii), conditions (a) and

(b) of Theorem 10 and the minimality of *g* 0 , we deduce that 2|*W*<sup>0</sup> 2 \ *V*2| + |*W*<sup>0</sup> 1 | ≤ |*V*1| + |*D*|. From the inequalities above we obtain

$$\begin{aligned} \prescript{s}{}{\arg}(G) &\leq \omega(\mathsf{g}') \\ &= |W\_1'| + 2|W\_2'| \\ &= |W\_1'| + 2|W\_2' \backslash V\_2| + 2|W\_2' \cap V\_2| \\ &\leq |V\_1| + |D| + 2|V\_2| \\ &= \gamma\_R(G) + \gamma(G). \end{aligned}$$

Therefore, the proof is complete.

*γ*

The bound above is tight. For instance, it is achieved for any graph *G* = *G*<sup>1</sup> + *G*<sup>2</sup> such that *γ*(*G*1) = 1. In this case, Theorem 3 results in *γ s tR*(*G*) = 3 = *γR*(*G*) + *γ*(*G*).

The following characterization is an immediate consequence of Theorem 11 and the well-known inequality *γR*(*G*) ≤ 2*γ*(*G*).

**Theorem 12.** *Let G be a graph with no isolated vertex. Then γ s tR*(*G*) = 3*γ*(*G*) *if and only if γ s tR*(*G*) = *γR*(*G*) + *γ*(*G*) *and γR*(*G*) = 2*γ*(*G*)*.*

From Theorem 4 and the fact that *γR*(*G*<sup>1</sup> *G*2) = 2*γ*(*G*<sup>1</sup> *G*2) = 2*n*(*G*1) we deduce that *γ s tR*(*G*<sup>1</sup> *G*2) = *γR*(*G*<sup>1</sup> *G*2) = 2*γ*(*G*<sup>1</sup> *G*2) for any graph *G*<sup>1</sup> with no isolated vertex and any nontrivial graph *G*2. This previous equality chain shows that the condition *γR*(*G*) = 2*γ*(*G*) is a necessary condition but is not sufficient to satisfy *γ s tR*(*G*) = 3*γ*(*G*).

We continue the study by providing a new upper bound, which improves the classical inequality *γ s tR*(*G*) ≤ 2*γt*(*G*). We need to introduce some concepts and tools. For any *γt*(*G*)-set *D*, let *D*<sup>∗</sup> ⊆ *D* be a set of minimum cardinality such that *D*<sup>∗</sup> ∈ D(*G*). Observe that *D*∗ is not necessarily a *γ*(*G*)-set. For instance, for the graph *G* given in Figure 2 we have that *γt*(*G*) = 4 and *γ*(*G*) = 3. However, the set *D* of black-colored vertices is the only *γt*(*G*)-set; moreover, the only dominating set that is a subset of *D* is *D* itself.

**Figure 2.** The set of black-colored vertices is the only *γt*(*G*)-set.

We define *KG*(*D*) = *D* \ *D*<sup>∗</sup> as the *kernel* of *D*. The maximum cardinality among all kernels *KG*(*D*) from all *γt*(*G*)-sets *D* is the *kernel* of *G* and it is denoted by *k*(*G*). For instance, *k*(*G*<sup>1</sup> *G*2) = 0 and also if *γ*(*G*<sup>1</sup> + *G*2) = 1, then *k*(*G*<sup>1</sup> + *G*2) = 1.

**Theorem 13.** *For any graph G with no isolated vertex,*

$$
\gamma\_{tR}^s(G) \le 2\gamma\_t(G) - k(G).
$$

**Proof.** Let *D* be a *γt*(*G*)-set such that *k*(*G*) = |*KG*(*D*)|. Let *D*<sup>∗</sup> ⊆ *D* ∩ D(*G*) be the set such that *KG*(*D*) = *D* \ *D*<sup>∗</sup> . Notice that the function *f*(*V*0, *V*1, *V*2), defined by *V*<sup>2</sup> = *D*<sup>∗</sup> , *V*<sup>1</sup> = *KG*(*D*) and *V*<sup>0</sup> = *V*(*G*) \ *D*, is a STRDF on *G*. Hence,

$$\begin{aligned} \prescript{s}{}{\prescript{}{t}}(G) &\leq \omega(f) \\ &= |V\_1| + 2|V\_2| \\ &= |K\_G(D)| + 2|D^\* \\ &= 2|D| - |K\_G(D)| \\ &= 2\gamma\_t(G) - k(G). \end{aligned}$$


*γ*

Therefore, the proof is complete.

*γ*

The following result provides a necessary condition for the graphs *G* satisfying *γ s tR*(*G*) = 2*γt*(*G*).

**Theorem 14.** *Let G be a graph of order at least three with no isolated vertex. If γ s tR*(*G*) = 2*γt*(*G*)*, then epn*(*v*, *D*) 6= ∅ *for every γt*(*G*)*-set D and v* ∈ *D.*

**Proof.** If there exist a *γt*(*G*)-set *D* and a vertex *v* ∈ *D* such that *epn*(*v*, *D*) = ∅, then |*KG*(*D*)| ≥ 1 because *D* \ {*v*} ∈ D(*G*). Hence, *k*(*G*) ≥ 1 and Theorem 13 results in *γ s tR*(*G*) < 2*γt*(*G*), which completes the proof.

The following results provide lower bounds for the strongly total Roman domination number in terms of order, maximum degree and total domination number of a graph.

**Theorem 15.** *For any graph G with every component of order at least three,*

$$
\gamma\_{tR}^s(G) \ge \gamma\_t(G) + \frac{\mathfrak{n}(G) - \gamma\_t(G)}{\Delta(G) - 1}.
$$

**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γ s tR*(*G*)-function. As *V*<sup>1</sup> ∪ *V*<sup>2</sup> ∈ D*t*(*G*), we deduce that

$$|V\_2| = \omega(f) - (|V\_1| + |V\_2|) \le \gamma\_{tR}^s(G) - \gamma\_t(G).$$

Now, it is easy to deduce that |*V*0| ≤ (∆(*G*) − 1)|*V*2| because *V*<sup>2</sup> ∈ D(*G*). Hence,

$$\begin{aligned} \prescript{s}{}{\scriptscriptstyle{(}\!| } (\mathcal{G}) &= |V\_1| + 2|V\_2| \\ &= n(\mathcal{G}) - |V\_0| + |V\_2| \\ &\ge n(\mathcal{G}) - (\Delta(\mathcal{G}) - 1)|V\_2| + |V\_2| \\ &= n(\mathcal{G}) - (\Delta(\mathcal{G}) - 2)|V\_2| \\ &\ge n(\mathcal{G}) - (\Delta(\mathcal{G}) - 2)(\gamma\_{IR}^{\mathcal{S}}(\mathcal{G}) - \gamma\_{\mathcal{I}}(\mathcal{G})) .\end{aligned}$$

Therefore, we deduce that *γ s tR*(*G*) <sup>≥</sup> *<sup>γ</sup>t*(*G*) + *<sup>n</sup>*(*G*)−*γt*(*G*) ∆(*G*)−1 , which completes the proof.

In order to show a class of graphs satisfying the equality in the previous bound, we consider the corona product graphs *K*<sup>2</sup> *H*. For these graphs we obtain that

$$\gamma\_{t\mathcal{K}}^s(\mathcal{K}\_2 \odot H) = 4 = \gamma\_t(\mathcal{K}\_2 \odot H) + \frac{n(\mathcal{K}\_2 \odot H) - \gamma\_t(\mathcal{K}\_2 \odot H)}{\Delta(\mathcal{K}\_2 \odot H) - 1} \lambda$$

because *γ s tR*(*K*<sup>2</sup> *H*) = 4 by Theorem 4, *γt*(*K*<sup>2</sup> *H*) = 2, *n*(*K*<sup>2</sup> *H*) = 2*n*(*H*) + 2 and ∆(*K*<sup>2</sup> *H*) = *n*(*H*) + 1.

In [15], the authors showed that *γtR*(*G*) ≥ 2*n*(*G*) ∆(*G*) for any graph *G* with no isolated vertex. The following result is a direct consequence of this previous inequality and Theorems 2, 5 and 15.

**Theorem 16.** *For any graph G with no isolated vertex,*

$$\gamma\_{tR}^s(G) \ge \left\lceil \frac{2n(G)}{\Delta(G)} \right\rceil$$

.

*Furthermore, if <sup>γ</sup>t*(*G*) = *<sup>n</sup>*(*G*) ∆(*G*) *, then the previous bound is achieved.*

The next theorem shows another relationship between our parameter and the order, maximum degree and total domination number of a graph. This result improves the bound given in the previous theorem whenever *γt*(*G*) ≥ 2*n*(*G*) ∆(*G*) .

**Theorem 17.** *For any graph G with no isolated vertex,*

$$
\gamma\_{tR}^s(G) \ge \left\lceil \frac{2n(G) + \gamma\_t(G)}{\Delta(G) + 1} \right\rceil.
$$

**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γ s tR*(*G*)-function. As *V*<sup>1</sup> ∪ *V*<sup>2</sup> ∈ D*t*(*G*), we deduce that

> *γt*(*G*) ≤ |*V*1| + |*V*2| = *ω*(*f*) − |*V*2| = *γ s tR*(*G*) − |*V*2|.

Now, notice that the following is the case:

$$\begin{aligned} \Delta(G)\gamma\_{IR}^s(G) &= \Delta(G)\omega(f) \\ &= \Delta(G)\sum\_{\mathbf{x}\in V(G)} f(\mathbf{x}) \\ &\ge \sum\_{\mathbf{x}\in V(G)} |N(\mathbf{x})|f(\mathbf{x}) \\ &= \sum\_{\mathbf{x}\in V(G)} \sum\_{\mathbf{u}\in N(\mathbf{x})} f(\mathbf{u}) \\ &\ge 2|V\_0| + 2|V\_1| + |V\_2| \\ &= 2n(G) - |V\_2|. \end{aligned}$$

From previous inequality chains we deduce the following:

$$2\mathfrak{n}(\mathcal{G}) + \gamma\_{\mathfrak{t}}(\mathcal{G}) \le \Delta(\mathcal{G})\gamma\_{\mathfrak{t}\mathcal{R}}^{s}(\mathcal{G}) + |V\_{2}| + \gamma\_{\mathfrak{t}\mathcal{R}}^{s}(\mathcal{G}) - |V\_{2}| = (\Delta(\mathcal{G}) + 1)\gamma\_{\mathfrak{t}\mathcal{R}}^{s}(\mathcal{G}).$$
 
$$\text{Therefore, } \gamma\_{\mathfrak{t}\mathcal{R}}^{s}(\mathcal{G}) \ge \left\lceil \frac{2\mathfrak{n}(\mathcal{G}) + \gamma\_{\mathfrak{t}}(\mathcal{G})}{\Delta(\mathcal{G}) + 1} \right\rceil \text{, as desired. } \square$$

The bound above is tight. For instance, it is achieved for any graph *G* such that ∆(*G*) = *n*(*G*) − 1.

#### **5. Conclusions and Open Problems**

In this article we introduced the concept of strongly total Roman domination number and showed that this parameter is an appropriate framework to study the total Roman domination number of lexicographic product graphs. Moreover, we obtained new tight bounds and provided exact formulas for some product graphs. As a consequence of this study, we showed that the problem of computing *γ s tR*(*G*) is NP-hard.

We next propose some open problems which we consider to be interesting:


$$\mathbf{(a)}\qquad\gamma\_{tR}^{s}(G)=\mathfrak{n}(G);$$

$$\mathbf{(b)}\qquad\gamma\_{tR}^s(G)=\gamma\_R(G)+\gamma(G);$$

$$\text{(c)}\qquad\gamma\_{tR}^{\ddot{s}}(G) = 2\gamma\_t(G) - k(G).$$

**Author Contributions:** The work was organized and led by A.C.M. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

