*Article* **Local Inclusive Distance Vertex Irregular Graphs**

**Kiki Ariyanti Sugeng 1,\*,†, Denny Riama Silaban 1,†, Martin Baˇca 2,† and Andrea Semaniˇcová-Fe ˇnovˇcíková 2,†**


**Abstract:** Let *G* = (*V*, *E*) be a simple graph. A vertex labeling *f* : *V*(*G*) → {1, 2, ... , *k*} is defined to be a local inclusive (respectively, non-inclusive) *d*-distance vertex irregular labeling of a graph *G* if for any two adjacent vertices *x*, *y* ∈ *V*(*G*) their weights are distinct, where the weight of a vertex *x* ∈ *V*(*G*) is the sum of all labels of vertices whose distance from *x* is at most *d* (respectively, at most *d* but at least 1). The minimum *k* for which there exists a local inclusive (respectively, non-inclusive) *d*-distance vertex irregular labeling of *G* is called the local inclusive (respectively, non-inclusive) *d*-distance vertex irregularity strength of *G*. In this paper, we present several basic results on the local inclusive *d*-distance vertex irregularity strength for *d* = 1 and determine the precise values of the corresponding graph invariant for certain families of graphs.

**Keywords:** (inclusive) distance vertex irregular labeling; local (inclusive) distance vertex irregular labeling

**MSC:** 05C15; 05C78

#### **1. Introduction**

All graphs considered in this paper are simple finite. We use *V*(*G*) for the vertex set and *E*(*G*) for the edge set of a graph *G*. The neighborhood *NG*(*x*) of a vertex *x* ∈ *V*(*G*) is the set of all vertices adjacent to *x*, which is a set of vertices whose distance from *x* is 1. Otherwise, *NG*[*x*] denotes the set of all neighbors of a vertex *x* ∈ *V*(*G*) including *x*, which is the set of vertices whose distance from *x* is at most 1. We are following the standard notation and the terminology presented in [1].

The notion of the irregularity strength was introduced by Chartrand et al. in [2]. For a given edge *k*-labeling *α* : *E*(*G*) → {1, 2, ... , *k*}, where *k* is a positive integer, the associated weight of a vertex *<sup>x</sup>* ∈ *<sup>V</sup>*(*G*) is *<sup>w</sup>α*(*x*) = <sup>∑</sup>*y*∈*NG*(*x*) *<sup>α</sup>*(*xy*). Such a labeling *<sup>α</sup>* is called *irregular* if *wα*(*x*) = *wα*(*y*) for every pair *x*, *y* of vertices of *G*. The smallest integer *k* for which an irregular labeling of *G* exists is known as the *irregularity strength* of *G*. This parameter has attracted much attention, see [3–5].

Inspired by irregularity strength and distance magic labeling defined in [6] and investigated in [7], Slamin [8] introduced the concept of a distance vertex irregular labeling of graphs. A *distance vertex irregular labeling* of a graph is a mapping *β* : *V*(*G*) → {1, 2, ... , *k*} such that the set of vertex weights consists of distinct numbers, where the weight of a vertex *<sup>x</sup>* ∈ *<sup>V</sup>*(*G*) under the labeling *<sup>β</sup>* is defined as *wtβ*(*x*) = <sup>∑</sup>*y*∈*NG*(*x*) *<sup>β</sup>*(*y*). The minimum *k* for which a graph *G* has a distance vertex irregular labeling is called the *distance vertex irregularity strength* of *G* and is denoted by dis(*G*).

In [8], Slamin determined the exact value of the distance vertex irregularity strength for complete graphs, paths, cycles and wheels, namely dis(*Kn*) = *n*, for *n* ≥ 3, dis(*Pn*) = *n*/2, for *n* ≥ 4, dis(*Cn*) = (*n* + 1)/2, for *n* ≡ 0, 1, 2, 3 (mod 8) and dis(*Wn*) =

**Citation:** Sugeng, K.A.; Silaban, D.R.; Baˇca, M.; Semaniˇcová-Fe ˇnovˇcíková, A. Local Inclusive Distance Vertex Irregular Graphs. *Mathematics* **2021**, *9*, 1673. https://doi.org/10.3390/ math9141673

Academic Editors: Janez Žerovnik and Darja Rupnik Poklukar

Received: 23 June 2021 Accepted: 13 July 2021 Published: 16 July 2021

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(*n* + 1)/2, for *n* ≡ 0, 1, 2, 5 (mod 8). Completed results for cycles and wheels are proved in [9].

Bong et al. [10] generalized the concept of a distance vertex irregular labeling to inclusive and non-inclusive *d*-distance vertex irregular labelings. The difference between inclusive and non-inclusive labeling depends on the way whether the vertex label is included in the vertex weight or not. The symbol *d* represents how far the neighborhood is considered. Thus, an inclusive (respectively, non-inclusive) *d*-distance vertex irregular labeling of a graph *G* is a mapping *β* such that the set of vertex weights consists of distinct numbers, where the weight of a vertex *x* ∈ *V*(*G*) is the sum of all labels of vertices whose distance from *x* is at most *d* (respectively, at most *d* but at least 1). The minimum *k* for which there exists an inclusive (respectively, non-inclusive) *d*-distance vertex irregular labeling of a graph *G* is called the *inclusive (respectively, non-inclusive) d-distance vertex irregularity strength* of *G*. The non-inclusive 1-distance vertex irregularity strength of a graph *G* is using Slamin's [8] terminology known as the distance vertex irregularity strength of *G*, denoted by dis(*G*). For the inclusive 1-distance vertex irregularity strength, we will use notation idis(*G*).

In [10] is determined the inclusive 1-distance vertex irregularity strength for paths *Pn*, *n* ≡ 0 (mod 3), stars, double stars *S*(*m*, *n*) with *m* ≤ *n*, a lower bound for caterpillars, cycles, and wheels. In [11] is established a lower bound of the inclusive 1-distance vertex irregularity strength for any graph and determined the exact value of this parameter for several families of graphs, namely for complete and complete bipartite graphs, paths, cycles, fans, and wheels. More results on triangular ladder and path for *d* ≥ 1 has been proved in [12,13].

Motivated by a distance vertex labeling [8], an irregular labeling [2] and a recent paper on a local antimagic labeling [14], we introduce in this paper the concept of local inclusive and local non-inclusive *d*-distance vertex irregular labelings.

A vertex labeling *f* : *V*(*G*) → {1, 2, ... , *k*} is defined to be a local *inclusive* (respectively, *non-inclusive*) *d-distance vertex irregular labeling* of a graph *G* if for any two adjacent vertices *x*, *y* ∈ *V*(*G*) their weights are distinct, where the weight of a vertex *x* ∈ *V*(*G*) is the sum of all labels of vertices whose distance from *x* is at most *d* (respectively, at most *d* but at least 1). The minimum *k* for which there exists a local inclusive (respectively, non-inclusive) *d*distance vertex irregular labeling of *G* is called the *local inclusive* (respectively, *non-inclusive*) *d-distance vertex irregularity strength* of *G* and denoted by lidis*d*(*G*) (respectively, ldis*d*(*G*)). If there is no such labeling for the graph *G* then the value of lidis*d*(*G*) is defined as ∞. In the case when *d* = 1 the index *d* can be omitted, thus lidis1(*G*) = lidis(*G*) (respectively, ldis1(*G*) = ldis(*G*)). In this paper, we only discuss the case for inclusive labeling with *d* = 1. Note that the concept of a local non-inclusive distance vertex irregular labeling has been introduced earlier in [15] with a different name. For more information about labeled graphs see [16].

In this paper, we present several basic results and some estimations on the local inclusive 1-distance vertex irregularity strength and determine the precise values of the corresponding graph invariant for several families of graphs.

#### **2. Basic Properties**

In the following observations, we give several basic properties of lidis(*G*). The first observation gives a relation between the local inclusive distance vertex irregularity strength, lidis(*G*), and the inclusive distance vertex irregularity strength, idis(*G*). The second and third observations give the requirement for giving the label of two vertices which have a common neighbor.

**Observation 1.** *For a graph G, it holds that* lidis(*G*) ≤ idis(*G*)*.*

**Observation 2.** *If there exists an edge uv in a graph G such that NG*(*u*) − {*v*} = *NG*(*v*) − {*u*}*, then for any local non-inclusive distance vertex irregular labeling f of a graph G holds f*(*u*) = *f*(*v*)*.* **Observation 3.** *If there exists an edge uv in a graph G such that NG*(*u*) − {*v*} = *NG*(*v*) − {*u*}*, then* lidis(*G*) = ∞*.*

The next theorem gives a sufficient and necessary condition for lidis(*G*) < ∞. Note that the graph *G* is not necessarily connected.

**Theorem 1.** *For a graph G, it holds that* lidis(*G*) = ∞ *if and only if there exists an edge uv* ∈ *E*(*G*) *such that NG*[*u*] = *NG*[*v*]*.*

**Proof.** If there exists an edge *uv* ∈ *E*(*G*) such that *NG*[*u*] = *NG*[*v*], then immediately follows Observation 3 and we obtain lidis(*G*) = ∞. On the other hand, if lidis(*G*) = ∞ then there exist at least two vertices *u* and *v* in *G* that have the same weight under any vertex labeling. It is only happened if *NG*[*u*] = *NG*[*v*].

Immediately from the previous theorem we obtain the following result.

**Corollary 1.** *If there exist two distinct vertices u*, *v in G such that* deg*G*(*u*) = deg*G*(*v*) = |*V*(*G*)| − 1*, then* lidis(*G*) = ∞*.*

Thus, for complete graphs we obtain

**Corollary 2.** *Let n be a positive integer. Then*

$$\text{lidis}(K\_n) = \begin{cases} 1, & \text{if } n = 1, \\ \infty, & \text{if } n \ge 2. \end{cases}$$

Now, we present a sufficient and necessary condition for lidis(*G*) = 1.

**Theorem 2.** *Let G be a graph. Then* lidis(*G*) = 1 *if and only if for every edge uv* ∈ *E*(*G*)*,* deg(*u*) = deg(*v*)*.*

**Proof.** Consider a labeling that assigns number 1 to every vertex of a graph *G*. Under this labeling, the weight of any vertex *v* in *G* is *wt*(*v*) = deg*G*(*v*) + 1. Thus, adjacent vertices can have distinct weights if and only if they have distinct degrees.

The chromatic number of a graph *G*, denoted by *χ*(*G*), is the smallest number of colors needed to color the vertices of *G* so that no two adjacent vertices share the same color, see [1]. The following result gives a trivial lower bound for the number of distinct induced vertex weights under any local inclusive distance vertex irregular labeling of a graph *G*.

**Theorem 3.** *For a graph G, the number of distinct induced vertex weights under any local inclusive distance vertex irregular labeling is at least χ*(*G*)*.*

#### **3. Local Inclusive Distance Vertex Irregularity Strength for Several Families of Graphs**

In this section, we provide the exact values of local inclusive distance vertex irregularity strengths of some standard graphs such as paths, cycles, complete bipartite graphs, complete multipartite graphs, and caterpillars. We also give results on several products of graphs, such as corona graphs, union graphs, and join product graphs.

**Theorem 4.** *Let Cn be a cycle on n vertices n* ≥ 3*. Then*

$$\text{lidis}(\mathbf{C}\_n) = \begin{cases} \infty, & \text{if } n = 3, \\ 2, & \text{if } n \text{ is even,} \\ 3, & \text{if } n \text{ is odd, } n \ge 5. \end{cases}$$

**Proof.** Let *V*(*Cn*) = {*vi* : *i* = 1, 2, ... , *n*} be the vertex set and let *E*(*Cn*) = {*vivi*+<sup>1</sup> : *i* = 1, 2, ... , *n* − 1}∪{*v*1*vn*} be the edge set of a cycle *Cn*. The lower bound for the local inclusive distance vertex irregularity strength of *Cn* follows from Theorem 3 as

$$\chi(\mathbf{C}\_n) = \begin{cases} \mathbf{3}, & \text{if } n \text{ is odd,} \\ \mathbf{2}, & \text{if } n \text{ is even.} \end{cases}$$

As *C*<sup>3</sup> is isomorphic to *K*<sup>3</sup> we use Corollary 2 in this case. For *n* even, we label the vertices of *Cn* as follows

$$f(v\_i) = \begin{cases} 1, & \text{if } i \text{ is odd,} \\ 2, & \text{if } i \text{ is even.} \end{cases}$$

Then, for the vertex weights we obtain

$$wt\_f(v\_i) = \begin{cases} 5, & \text{if } i \text{ is odd,} \\ 4, & \text{if } i \text{ is even.} \end{cases}$$

Thus, for *n* even we obtain lidis(*Cn*) = 2.

For *n* = 5, we label the vertices such that *f*(*v*1) = *f*(*v*2) = 1, *f*(*v*3) = 3 and *f*(*v*4) = *f*(*v*5) = 2. Then, *wtf*(*v*1) = 4, *wtf*(*v*2) = *wtf*(*v*5) = 5, *wtf*(*v*3) = 6 and *wtf*(*v*4) = 7. Thus, lidis(*C*5) = 3.

For *n* odd, *n* ≥ 7, the vertices are labeled in the following way

$$f(v\_i) = \begin{cases} 1, & \text{if } i \text{ is odd, } 1 \le i \le n - 4, \\ 2, & \text{if } i \text{ is even, } 2 \le i \le n - 3, \\ 3, & \text{if } i = n - 2, n - 1, n. \end{cases}$$

The weights of vertices are

$$wt\_f(v\_i) = \begin{cases} 6, & \text{if } i = 1, n - 3, \\ 5, & \text{if } i \text{ is odd, } 3 \le i \le n - 4, \\ 4, & \text{if } i \text{ is even, } 2 \le i \le n - 5, \\ 8, & \text{if } i = n - 2, \\ 9, & \text{if } i = n - 1, \\ 7, & \text{if } i = n. \end{cases}$$

As adjacent vertices have distinct weights we obtain lidis(*Cn*) = 3 for *n* odd. The above explanation concludes the proof.

**Corollary 3.** *Let Pn be a path on n vertices n* ≥ 2*. Then*

$$\text{lidis}(P\_n) = \begin{cases} \infty, & \text{if } n = 2, \\ 2, & \text{if } n \ge 3. \end{cases}$$

**Proof.** Let *V*(*Pn*) = {*vi* : *i* = 1, 2, ... , *n*} be the vertex set and let *E*(*Pn*) = {*vivi*+<sup>1</sup> : *i* = 1, 2, . . . , *n* − 1} be the edge set of a path *Pn*. The result for *n* = 2 follows from Corollary 2.

For *n* ≥ 3, according to Theorem 3, the lidis(*Pn*) should be more than one. Using the vertex labels for *n* even as in Theorem 4 and the corresponding vertex weights are

$$wt\_f(v\_i) = \begin{cases} \mathfrak{Z}, & \text{if } i = 1, n\_\prime \\ 4, & \text{if } i \text{ is even, } i \neq n\_\prime \\ 5, & \text{if } i \text{ is odd, } i \neq 1 \text{ and } i \neq n\_\prime \end{cases}$$

Thus, lidis(*Pn*) = 2.

The following result deals with complete multipartite graphs.

**Theorem 5.** *Let Kn*1,*n*2,...,*nm be a complete multipartite graph, ni* ≥ 1*, i* = 1, 2, ... , *m, m* ≥ 2*. Then,*

$$\text{lidis}(K\_{n\_1, n\_2, \dots, n\_m}) = \begin{cases} \infty, & \text{if } 1 = n\_1 = n\_2 \\ 1, & \text{if } n\_1 < n\_2 < \dots < n\_m \\ m, & \text{if } 2 \le n\_1 = n\_2 = \dots = n\_m \end{cases}$$

**Proof.** Let us denote the vertices in the independent set *Vi*, *i* = 1, 2, ... , *m* of a complete multipartite graph *Kn*1,*n*2,...,*nm* by symbols *v*<sup>1</sup> *<sup>i</sup>* , *<sup>v</sup>*<sup>2</sup> *<sup>i</sup>* ,..., *<sup>v</sup>ni i* .

If 1 = *n*<sup>1</sup> = *n*2, then the vertices *v*<sup>1</sup> <sup>1</sup> and *<sup>v</sup>*<sup>1</sup> <sup>2</sup> have the same degrees

$$\deg(v\_1^1) = \deg(v\_2^1) = \sum\_{j=3}^m n\_j + 1 = |V(\mathcal{K}\_{\mathcal{W}\_1, \mathcal{W}\_2, \dots, \mathcal{W}\_m})| - 1$$

and thus, by Corollary 1 we obtain lidis(*Kn*1,*n*2,...,*nm* ) = ∞.

If *n*<sup>1</sup> < *n*<sup>2</sup> < ··· < *nm*, then all adjacent vertices have distinct degrees. More precisely, the degree of a vertex *v j i* , *i* = 1, 2, ... , *m*, *j* = 1, 2, ... , *ni* is deg(*v j i* ) = ∑*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *nj* − *ni* + 1. Thus, by Theorem 2, we obtain lidis(*Kn*1,*n*2,...,*nm* ) = 1.

If 2 ≤ *n*<sup>1</sup> = *n*<sup>2</sup> = ··· = *nm* = *n* consider a vertex labeling *f* of *Kn*1,*n*2,...,*nm* defined such that

$$f(v\_i^j) = i$$

for *i* = 1, 2, . . . , *m*, *j* = 1, 2, . . . , *n* and the corresponding vertex weights are

$$wt\_f(v\_i^j) = \frac{nm(m+1)}{2} - (n-1)i.$$

Thus, all adjacent vertices have distinct weights. Thus, lidis(*Kn*1,*n*2,...,*nm* ) ≤ *m*. Using mathematical induction, it is not complicated to show that lidis(*Kn*1,*n*2,...,*nm* ) ≥ *m*. This concludes the proof.

The following corollary gives the exact value of the studied parameter for complete bipartite graphs.

**Corollary 4.** *Let Km*,*n,* 1 ≤ *m* ≤ *n, be a complete bipartite graph. Then*

$$\text{lidis}(K\_{m,n}) = \begin{cases} \infty, & \text{if } m = n = 1, \\ 2, & \text{if } m = n \ge 2, \\ 1, & \text{if } m \ne n. \end{cases}$$

The *corona product* of *G* and *H* is the graph *G H* obtained by taking one copy of *G*, called the center graph along with |*V*(*G*)| copies of *H*, called the outer graph, and making the *i*th vertex of *G* adjacent to every vertex of the *i*th copy of *H*, where 1 ≤ *i* ≤ |*V*(*G*)|. For arbitrary graphs *G*, we can prove the following result.

**Theorem 6.** *Let r be a positive integer. Then, for r* ≥ 2 *holds*

$$\text{lidis}(G \odot \overline{K\_r}) \le \text{lidis}(G).$$

*Moreover, if G is a graph with no component of order 1 then also* lidis(*G K*1) ≤ lidis(*G*)*.*

**Proof.** If lidis(*G*) = ∞ then by Theorem 1 there exists at least one edge *uv* ∈ *E*(*G*) such that *NG*[*u*] = *NG*[*v*]. However, as for *r* ≥ 2 or for *r* = 1 if *G* has no component of order 1, in *G Kr* all vertices have distinct closed neighborhood and thus lidis(*G Kr*) < ∞.

Now, consider that lidis(*G*) < ∞ and let *f* be a local inclusive distance vertex irregular labeling of *G*. We define a labeling *g* of *G Kr* such that

$$\begin{aligned} g(v) &= f(v), \quad &\text{if } v \in V(G),\\ g(v) &= 1, \quad &\text{if } \deg\_{G \subset \overline{\mathbb{K}\_r}}(v) = 1. \end{aligned}$$

For the vertex weights, we obtain

$$\begin{aligned} wt\_{\mathcal{S}}(v) &= wt\_f(v) + r, & \text{if } v \in V(G), \\ wt\_{\mathcal{S}}(v) &= 1 + f(u), & \text{if } \deg\_{G \odot \overline{K\_r}}(v) &= 1 \text{ and } \mu v \in E(G \odot \overline{K\_r}). \end{aligned}$$

Evidently, for *r* ≥ 2 or for *r* = 1 if *G* has no component of order 1, i.e., deg*G*(*v*) ≥ 1 for every *v* ∈ *V*(*G*), we obtain that under the labeling *g* the vertex weights of adjacent vertices are different.

Moreover, we can prove that the parameter lidis(*G Kr*) is finite except the case when *G Kr* contains a component isomorphic to *K*2.

**Theorem 7.** *Let r be a positive integer. Then,*

$$\text{lidis}(G \odot \overline{K\_r}) \le |V(G)|$$

*except the case when r* = 1 *and the graph G contains a component of order 1.*

**Proof.** Let us denote the vertices of a graph *<sup>G</sup>* by symbols *<sup>v</sup>*1, *<sup>v</sup>*2, ... , *<sup>v</sup>*|*V*(*G*)<sup>|</sup> such that for every *i* = 1, 2, . . . , |*V*(*G*)| − 1 holds

$$
\deg\_G(v\_i) \le \deg\_G(v\_{i+1})
$$

and let *v j i* , *j* = 1, 2, ... ,*r* be the vertices of degree 1 adjacent to *vi*, *i* = 1, 2, ... , |*V*(*G*)|, in *G Kr*. Now, we define a labeling *f* that assigns 1 to every vertex of *G*. Thus, for every *i* = 1, 2, . . . , |*V*(*G*)|

$$wt\_f(v\_i) = \deg\_G(v\_i) + 1.$$

We extend the labeling *f* of the graph *G* to the labeling *g* of the graph *G Kr* in the following way

$$\begin{aligned} g(v\_i) &= f(v\_i), & \text{if } i &= 1, 2, \dots, \left| V(\mathbf{G}) \right|, \\ g(v\_i^j) &= i, & \text{if } i &= 1, 2, \dots, \left| V(\mathbf{G}) \right|, j = 1, 2, \dots, r. \end{aligned}$$

The induced vertex weights are

$$\begin{aligned} wt\_{\mathcal{S}}(v\_i) &= \deg\_{\mathcal{G}}(v\_i) + 1 + ri\_i & \quad \text{if } i = 1, 2, \dots, |V(\mathcal{G})|,\\ wt\_{\mathcal{S}}(v\_i^j) &= 1 + i\_i & \quad \text{if } i = 1, 2, \dots, |V(\mathcal{G})|, j = 1, 2, \dots, r. \end{aligned}$$

For *r* ≥ 2 and for *r* = 1 if the graph *G* has no component of order 1, i.e., deg(*vi*) ≥ 1 for every *i* = 1, 2, ... , |*V*(*G*)|, we obtain that all adjacent vertices have distinct weights.

Note that the upper bound in the previous theorem is tight, since lidis(*Kn K*1) = *n*. Immediately, from Theorem 2, we have the following result

**Theorem 8.** *For r* ≥ 2 *it holds* lidis(*G Kr*) = 1 *if and only if* lidis(*G*) = 1*.*

*Moreover, when G has no component of order 1 then* lidis(*G K*1) = 1 *if and only if* lidis(*G*) = 1*.*

Now, we present results for corona product of paths, cycles, and complete graphs with totally disconnected graph *Kr*, *r* ≥ 1. Combining Theorems 3 and 6, we obtain

**Theorem 9.** *Let Pn be a path on n vertices n* ≥ 2 *and let r be a positive integer. Then*

$$\text{lidis}(P\_n \odot \overline{K\_r}) = 2.$$

**Theorem 10.** *Let Cn be a cycle on n vertices n* ≥ 3 *and let r be a positive integer. Then*

$$\text{lidis}(\mathbb{C}\_n \odot \overline{K\_r}) = \begin{cases} \mathfrak{Z}\_\prime & \text{if } n = 3 \text{ and } r = 1 \\ \mathfrak{Z}\_\prime & \text{otherwise.} \end{cases}$$

**Proof.** Let

$$V(\mathbb{C}\_{\mathbb{H}} \odot \overline{K\_{\mathbb{r}}}) = \{v\_{\mathbb{i}} : i = 1, 2, \dots, n\} \cup \{v\_{\mathbb{i}}^{j} : i = 1, 2, \dots, n; j = 1, 2, \dots, r\}$$

be the vertex set and let

$$E(\mathbb{C}\_n \odot \overline{K\_r}) = \{v\_i v\_{i+1} : i = 1, 2, \dots, n - 1\} \cup \{v\_1 v\_n\}$$

$$\cup \{v\_i v\_j^j : i = 1, 2, \dots, n; j = 1, 2, \dots, r\}$$

be the edge set of *Cn Kr*.

For even *n* the result follows from Theorems 4 and 6. For *n* = 3 and *r* = 1 consider the labeling illustrated on Figure 1.

**Figure 1.** A local inclusive distance vertex irregular labeling of *C*<sup>3</sup> *K*1.

For odd *n* and (*n*,*r*) = (3, 1), we define a vertex labeling *f* of *Cn Kr* such that

$$\begin{aligned} f(v\_i) &= \begin{cases} 2, & \text{for } i = 1, \\ 1, & \text{for } i = 2, 3, \dots, n, \end{cases} \\ f(v\_i^j) &= \begin{cases} 2, & \text{for } i = 2, 4, \dots, n - 1, n \text{ and } j = 1, 2, \\ 1, & \text{otherwise.} \end{cases} \end{aligned}$$

The weights of vertices of degree *r* + 2 are

$$wt\_f(v\_i) = \begin{cases} r+3, & \text{for } i = 3, 5, \dots, n-2, \\ r+4, & \text{for } i = 1, 4, 6, \dots, n-1, \\ r+5, & \text{for } i = 2, n. \end{cases}$$

As the weights of vertices of degree one are either 2 or 3, we obtain that adjacent vertices have distinct weights.

**Theorem 11.** *Let n*,*r be positive integers. Then*

$$\text{lidis}(K\_{\mathfrak{n}} \odot \overline{K\_{\mathfrak{r}}}) = \begin{cases} \infty, & \text{if } \mathfrak{n} = 1, r = 1, \\ 1 + \left\lceil \frac{\mathfrak{n} - 1}{r} \right\rceil, & \text{otherwise}. \end{cases}$$

**Proof.** As the graph *K*<sup>1</sup> *K*<sup>1</sup> is isomorphic to the complete graph *K*<sup>2</sup> we use Corollary 2 in this case.

Let (*n*,*r*) = (1, 1). Let the vertex set and the edge set of *Kn Kr* be the following

$$V(K\_{\boldsymbol{\eta}} \odot \overline{K\_{\boldsymbol{r}}}) = \{v\_{i\boldsymbol{\prime}}v\_{i\boldsymbol{\prime}}^{j} : i = 1, 2, \ldots, n; \; j = 1, 2, \ldots, r\},$$

$$E(K\_{\boldsymbol{\eta}} \odot \overline{K\_{\boldsymbol{r}}}) = \{v\_{i}v\_{j} : i = 1, 2, \ldots, n - 1; \; j = i + 1, i + 2, \ldots, n\},$$

$$\cup \{v\_{i}v\_{j}^{j} : i = 1, 2, \ldots, n; \; j = 1, 2, \ldots, r\}.$$

We define a vertex labeling *f* of *Kn Kr* such that

$$\begin{aligned} f(v\_i) &= 1 + \left\lceil \frac{n-1}{r} \right\rceil, \quad \text{if } i = 1, 2, \dots, n, \\ f(v\_i^j) &= \begin{cases} 1 + \left\lceil \frac{i-1}{r} \right\rceil, \quad \text{if } i = 1, 2, \dots, n, j = 1, 2, \dots, A\_i, \\ 1 + \left\lfloor \frac{i-1}{r} \right\rfloor, \quad \text{if } i = 1, 2, \dots, n, j = A\_i + 1, A\_i + 2, \dots, r\_r \end{cases} \end{aligned}$$

where for every *i* = 1, 2, . . . , *n* the parameter *Ai*, 1 ≤ *Ai* ≤ *r*, is defined such that

$$i - 1 \equiv A\_i \pmod{r}.$$

For the vertex weights, we obtain

$$\begin{split} wt\_f(v\_i) &= n(1 + \left\lceil \frac{n-1}{r} \right\rceil) + r + i - 1, \quad \text{if } i = 1, 2, \dots, n, \\ wt\_f(v\_i^j) &= \begin{cases} \left\lceil \frac{n-1}{r} \right\rceil + 2 + \left\lceil \frac{i-1}{r} \right\rceil, \quad \text{if } i = 1, 2, \dots, n, j = 1, 2, \dots, A\_{i\prime} \\ \left\lceil \frac{n-1}{r} \right\rceil + 2 + \left\lfloor \frac{i-1}{r} \right\rfloor, \quad \text{if } i = 1, 2, \dots, n, j = A\_{\bar{i}} + 1, A\_{\bar{i}} + 2, \dots, r. \end{cases} \end{split}$$

Evidently adjacent vertices have distinct weights. Thus, as the maximal vertex label is 1 + (*n* − 1)/*r*, the proof is completed.

A *caterpillar* is a graph derived from a path by hanging any number of leaves from the vertices of the path. We denote the caterpillar as *Sn*1,*n*2,...,*nr*, where the vertex set is *V*(*Sn*1,*n*2,...,*nr* ) = {*ci* : 1 ≤ *i* ≤ *r*} ∪ *r i*=1{*u<sup>j</sup> <sup>i</sup>* : 1 ≤ *j* ≤ *ni*}, and the edge set is *E*(*Sn*1,*n*2,...*nr* ) = {*cici*+<sup>1</sup> : 1 ≤ *i* ≤ *r* − 1} ∪ *r i*=1{*ciu<sup>j</sup> <sup>i</sup>* : 1 ≤ *j* ≤ *ni*}.

**Theorem 12.** *For every caterpillar Sn*1,*n*2,...,*nr with at least 3 vertices holds* lidis(*Sn*1,*n*2,...,*nr* ) ≤ 2*.*

**Proof.** For a regular caterpillar, thus the case *n*<sup>1</sup> = *n*<sup>2</sup> = ... = *nr* = *n*, using Theorem 9, we obtain that lidis(*Sn*,*n*,...,*n*) = 2.

For the other cases, label the vertices of a caterpillar *Sn*1,*n*2,...,*nr* using the following algorithm.

	- Then the weights of vertices *ci*, *i* = 1, 2, ... ,*r* are deg(*ci*) and all vertices of degree 1 have weight 2.

After a finite number of steps, the algorithm stops and the weights of the vertices are always different from the weights of their neighbors.

A similar algorithm can be used to obtain a result for closed caterpillars, which are graphs where the removal of all pendant vertices gives a cycle. We denote the closed caterpillar as *CSn*1,*n*2,...,*nr*, where the vertex set is *V*(*CSn*1,*n*2,...,*nr* ) = {*ci* : 1 ≤ *i* ≤ *r*} ∪ *r i*=1{*u<sup>j</sup> <sup>i</sup>* : 1 ≤ *j* ≤ *ni*}, and the edge set is *E*(*CSn*1,*n*2,...*nr* ) = {*cici*+<sup>1</sup> : 1 ≤ *i* ≤ *r* − 1} ∪ {*c*1*cr*} ∪ *r i*=1{*ciu<sup>j</sup> <sup>i</sup>* : 1 ≤ *j* ≤ *ni*}.

**Theorem 13.** *For closed caterpillar CSn*1,*n*2,...,*nr holds*

$$\text{lidis}(\text{CS}\_{\text{lt}\_{1},\text{n}\_{2},\dots,\text{n}\_{r}}) = \begin{cases} \infty, & \text{if } r = 3 \text{ and } \{n\_{1}, n\_{2}, n\_{3}\} = \{n, 0, 0\}, \text{ where } n \ge 0, \\ \ $, & \text{if } r = 3 \text{ and } (n\_{1}, n\_{2}, n\_{3}) = (1, 1, 1), \\ \$ , & \text{if } r = 3 + 6k, k \ge 1 \text{ and } \{n\_{1}, n\_{2}, \dots, n\_{r}\} = \{1, 0, \dots, 0\}, \\ \le 2, & \text{otherwise.} \end{cases}$$

The proof of the next result for the disjoint union of graphs, follows from the fact that there are no edges between the distinct components.

**Theorem 14.** *Let Gi, i* = 1, 2, . . . , *m be arbitrary graphs. Then*

$$\text{lidis}\left(\bigcup\_{i=1}^{m}\mathcal{G}\_{i}\right) = \max\{\text{lidis}(\mathcal{G}\_{i}) : i = 1, 2, \dots, m\}.$$

Immediately from the previous theorem, we obtain the following result.

**Corollary 5.** *Let n be a non-negative integer and let G be a graph. Then,* lidis(*G* ∪ *nK*1) = lidis(*G*)*.*

The *join G* ⊕ *H* of the disjoint graphs *G* and *H* is the graph *G* ∪ *H* together with all the edges joining vertices of *V*(*G*) and vertices of *V*(*H*). Let Δ(*G*) denote the maximal degree of the graph *G*.

**Theorem 15.** *For any graph G holds*

$$\text{lidis}(G \oplus K\_1) = \begin{cases} \infty, & \text{if } \Delta(G) = |V(G)| - 1, \\ \text{lidis}(G), & \text{if } \Delta(G) < |V(G)| - 1. \end{cases}$$

**Proof.** Let *w* be the vertex of *K*1. In a graph *G* ⊕ *K*<sup>1</sup> the vertex *w* is adjacent to all vertices in *G* we immediately get that lidis(*G* ⊕ *K*1) ≥ lidis(*G*).

If Δ(*G*) = |*V*(*G*)| − 1 then in *G* ⊕ *K*<sup>1</sup> there are at least two vertices of degree |*V*(*G*)| = |*V*(*G* ⊕ *K*1)| − 1 and thus by Corollary 1 we have lidis(*G* ⊕ *K*1) = ∞.

Let Δ(*G*) < |*V*(*G*)| − 1. If lidis(*G*) = ∞ then by Theorem 1 there exists at least two vertices, say *u* and *v* in *G* such that *NG*[*u*] = *NG*[*v*]. However, these vertices have the same closed neighborhood also in the graph *G* ⊕ *K*<sup>1</sup> as

$$N\_{\mathbb{G}\oplus\mathcal{K}\_1}[\mathfrak{u}] = N\_{\mathbb{G}}[\mathfrak{u}] \cup \{\mathcal{w}\} = N\_{\mathbb{G}}[\mathfrak{v}] \cup \{\mathcal{w}\} = N\_{\mathbb{G}\oplus\mathcal{K}\_1}[\mathfrak{v}].$$

However, this implies that

$$\text{lidis}(G \oplus K\_1) = \infty = \text{lidis}(G).$$

Now, consider that lidis(*G*) < ∞ and let *f* be a corresponding local inclusive distance vertex irregular graph of *G*. We define a labeling *g* of *G* ⊕ *K*<sup>1</sup> in the following way

$$g(v) = \begin{cases} 1, & \text{if } v = w\_r \\ f(v), & \text{if } v \in V(G). \end{cases}$$

The induced vertex weights are

$$wt\_{\mathcal{S}}(v) = \begin{cases} \sum\_{\substack{\boldsymbol{v} \in V(\mathcal{G}) \\ w\boldsymbol{t}\_f(\boldsymbol{v}) + 1\_\prime}} f(\boldsymbol{u}) + 1\_\prime & \text{if } v = w\_\prime \\ w\boldsymbol{t}\_f(v) + 1\_\prime & \text{if } v \in V(\mathcal{G}). \end{cases}$$

As Δ(*G*) < |*V*(*G*)| − 1 we get that for any vertex *v* ∈ *V*(*G*) is

$$wt\_f(\upsilon) = \sum\_{\mu \in N\_G(\upsilon)} f(\mu) < \sum\_{\mu \in V(G)} f(\mu).$$

Thus, all adjacent vertices have distinct weights. This means that *g* is a local inclusive distance vertex irregular labeling of *G* ⊕ *K*1. As vertex *w* is adjacent to every vertex in *G* we get lidis(*G* ⊕ *K*1) = lidis(*G*) in this case. This concludes the proof.

The graph in the previous theorem is not necessarily connected.

**Theorem 16.** *Let Gi, i* = 1, 2, . . . , *m, m* ≥ 2 *be arbitrary graphs. Then*

$$\text{lidis}\left(\left(\bigcup\_{i=1}^{m} G\_i\right) \oplus K\_1\right) = \max\{\text{lidis}(G\_i) : i = 1, 2, \dots, m\}.$$

**Proof.** The proof follows from Theorems 14 and 15.

A *wheel Wn* with *n* spokes is isomorphic to the graph *Cn* ⊕ *K*1. A *fan graph Fn* is isomorphic to the graph *Pn* ⊕ *K*1, while a *generalized fan graph* is isomorphic to the graph *kPn* ⊕ *K*1. The following results are immediate corollaries of the previous theorems.

**Corollary 6.** *Let Wn be a wheel on n* + 1 *vertices n* ≥ 3*. Then*

$$\text{lidis}(W\_{\mathbb{N}}) = \begin{cases} \infty, & \text{if } n = 3, \\ 2, & \text{if } n \text{ is even,} \\ 3, & \text{if } n \text{ is odd, } n \ge 5. \end{cases}$$

**Corollary 7.** *Let Fn be a fan on n* + 1 *vertices n* ≥ 2*. Then*

$$\text{lidis}(F\_n) = \begin{cases} \infty, & \text{if } n = 2, \\ 2, & \text{if } n \ge 3. \end{cases}$$

**Corollary 8.** *Let kPn* ⊕ *K*<sup>1</sup> *be a generalized fan graph, k*, *n* ≥ 2*. Then*

$$\text{lidis}(kP\_n \oplus \mathcal{K}\_1) = 2.$$

If lidis(*G*) = ∞ then by Theorem 1 there exist at least two vertices, say *u* and *v* in *G* such that they have the same closed neighborhood *NG*[*u*] = *NG*[*v*]. Thus, we immediately get

$$\begin{aligned} \mathcal{N}\_{\mathcal{G}\oplus\overline{\mathcal{K}\_r}}[\iota] &= \mathcal{N}\_{\mathcal{G}}[\iota] \cup \{w\_i : i = 1, 2, \dots, r\} \\ &= \mathcal{N}\_{\mathcal{G}}[v] \cup \{w\_i : i = 1, 2, \dots, r\} = \mathcal{N}\_{\mathcal{G}\oplus\overline{\mathcal{K}\_r}}[v]\_{\mathcal{H}} \end{aligned}$$

where *wi*, *i* = 1, 2, ... ,*r*, are the vertices of *Kr*. Thus, lidis(*G* ⊕ *Kr*) = ∞ for every positive integer *r*. Now we will deal with the case when lidis(*G*) < ∞ and *r* ≥ 2.

**Theorem 17.** *Let r* ≥ 2 *be a positive integer and let G be not isomorphic to a totally disconnected graph. If* lidis(*G*) < ∞ *and r* ≥ |*V*(*G*)| · lidis(*G*) *then* lidis(*G* ⊕ *Kr*) = lidis(*G*)*.*

**Proof.** Let us denote the vertices *Kr* by the symbols *wi*, *i* = 1, 2, ... ,*r* and let *r* ≥ 2. Thus, *V*(*G* ⊕ *Kr*) = *V*(*G*) ∪ {*wi* : *i* = 1, 2, ... ,*r*}. In a graph *G* ⊕ *Kr* all the vertices *wi*, *i* = 1, 2, ... ,*r* are adjacent to all vertices in *G* thus we immediately get that lidis(*G* ⊕ *Kr*) ≥ lidis(*G*).

Let lidis(*G*) < ∞ and let *f* be a corresponding local inclusive distance vertex irregular labeling of *G*. We define a labeling *g* of *G* ⊕ *Kr* in the following way

$$g(v) = \begin{cases} 1, & \text{if } v = w\_{\text{i}} \text{ } i = 1, 2, \dots, r\_{\text{i}}, \\ f(v), & \text{if } v \in V(G). \end{cases}$$

Then, the vertex weights are

$$wt\_{\mathcal{S}}(v) = \begin{cases} \sum\_{\substack{\boldsymbol{f} \in V(\mathcal{G}) \\ \boldsymbol{w}t\_{\boldsymbol{f}}(\boldsymbol{v}) + \boldsymbol{r}\_{\boldsymbol{\nu}}}} \text{if } \boldsymbol{v} = w\_{\boldsymbol{i}\boldsymbol{\nu}} \text{ i } = 1, 2, \dots, r\_{\boldsymbol{\nu}} \\ \boldsymbol{w}t\_{\boldsymbol{f}}(\boldsymbol{v}) + \boldsymbol{r}\_{\boldsymbol{\nu}} & \text{if } \boldsymbol{v} \in V(\mathcal{G}). \end{cases}$$

Evidently, under the labeling *g*, all adjacent vertices in *V*(*G*) have distinct weights. We need also to prove that no vertex in *V*(*G*) has the same weight as in *V*(*Kr*). Consider that

$$r \ge |V(G)| \cdot \text{lidis}(G).$$

As *G* is not isomorphic to a totally disconnected graph then for the weight of any vertex *v* in *V*(*G*) we have

$$wt\_{\mathcal{S}}(v) = wt\_f(v) + r \ge 1 + |V(G)| \cdot \text{lidis}(G) > 1 + \sum\_{u \in V(G)} f(u) = wt\_{\mathcal{S}}(w\_l).$$

for every *i* = 1, 2, ... ,*r*. Thus, *g* is a local inclusive distance vertex irregular graph of *G* ⊕ *Kr* and hence lidis(*G* ⊕ *Kr*) ≤ lidis(*G*).

Note that for small *r* the previous theorem is not necessarily true. Consider the graph *G* illustrated on Figure 2, evidently lidis(*G*) = 1. However, lidis(*G* ⊕ *K*3) = 2.

**Figure 2.** A local inclusive distance vertex irregular labeling of a graph *G*.

#### **4. Conclusions**

In this paper, we introduced the local inclusive distance vertex irregularity strength of graphs and gave some basic results and also some constructions of the feasible labelings for several families of graphs. We still have some open problems and conjecture as follows:

**Problem 1.** *Find* lidis(*Kn*1,*n*2,...,*nm* ) *for general case, which is for the case n*<sup>1</sup> ≤ *n*<sup>2</sup> ≤···≤ *nm, where m* > 2*.*

**Problem 2.** *Characterize graphs for which* lidis(*G Kr*) = lidis(*G*)*.*

**Conjecture 1.** *For arbitrary tree T with T* = *K*2*,* lidis(*T*) = 1 *or* 2*.*

**Author Contributions:** Conceptualization, K.A.S., D.R.S., M.B. and A.S.-F.; methodology, K.A.S., D.R.S., M.B. and A.S.-F.; validation, K.A.S., D.R.S., M.B. and A.S.-F.; investigation, K.A.S., D.R.S., M.B. and A.S.-F.; resources, K.A.S., D.R.S., M.B. and A.S.-F.; writing—original draft preparation, K.A.S., and A.S.-F.; writing—review and editing, K.A.S., D.R.S., M.B. and A.S.-F.; supervision, K.A.S. and A.S.-F.; project administration, K.A.S., M.B. and A.S.-F.; funding acquisition, K.A.S., D.R.S., M.B. and A.S.-F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has supported by PUTI KI-Universitas Indonesia 2020 Research Grant No. NKB-779/UN2.RST/HKP.05.00/2020. This work was also supported by the Slovak Research and Development Agency under the contract No. APVV-19-0153 and by VEGA 1/0233/18.

**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**


**Abel Cabrera Martínez 1, Juan C. Hernández-Gómez <sup>2</sup> and José M. Sigarreta 2,\***


**Abstract:** Domination theory is a well-established topic in graph theory, as well as one of the most active research areas. Interest in this area is partly explained by its diversity of applications to real-world problems, such as facility location problems, computer and social networks, monitoring communication, coding theory, and algorithm design, among others. In the last two decades, the functions defined on graphs have attracted the attention of several researchers. The Romandominating functions and their variants are one of the main attractions. This paper is a contribution to the Roman domination theory in graphs. In particular, we provide some interesting properties and relationships between one of its variants: the quasi-total Roman domination in graphs.

**Keywords:** quasi-total Roman domination; total Roman domination; Roman domination

#### **1. Introduction**

Domination in graphs was first defined as a graph-theoretical concept in 1958. This area has attracted the attention of many researchers due to its diversity of applications to real-world problems, such as problems with the location of facilities, computing and social networks, communication monitoring, coding theory, and algorithm design, among others. In that regard, this topic has experienced rapid growth, resulting in over 5000 papers being published. We refer to [1,2] for theoretical results and practical applications.

Given a graph *G*, a *dominating set* is a subset *D* ⊆ *V*(*G*) of vertices, such that every vertex not in *D* is adjacent to at least one vertex in *D*. The minimum cardinality among all dominating sets of *G* is called the *domination number* of *G*. The number of works, results and open problems that exist on this parameter and its variants provide a very wide range of work directions to consider, which come from their very theoretical aspects to a significant number of practical applications, passing through a large number of relationships and connections between some invariants of graph theory itself.

In the last two decades, the interest in research concerning dominating functions in graphs has increased. One of the reasons for this is that dominating functions generalize the concept of dominating sets. In particular, the Roman dominating functions (defined in [3], due to historical reasons arising from the ancient Roman Empire and described in [4,5]), and their variants, are one of the main attractions. At present, more than 300 papers have been published on this topic.

In 2019, Cabrera García et al. [6] defined and began the study of an interesting variant of Roman-dominating functions: the quasi-total Roman-dominating functions. This paper deals precisely with this style of domination, and our goal is to continue with the study of this novel parameter in graphs.

#### *Definitions, Notation and Organization of the Paper*

We begin this subsection by stating the main basic terminology which will be used in the whole work. Let *G* = (*V*(*G*), *E*(*G*)) be a simple graph with no isolated vertex. Given

**Citation:** Martínez, A.C.; Hernández-Gómez, J.C.; Sigarreta, J.M. On the Quasi-Total Roman Domination Number of Graphs. *Mathematics* **2021**, *9*, 2823. https:// doi.org/10.3390/math9212823

Academic Editor: Frank Werner

Received: 28 September 2021 Accepted: 2 November 2021 Published: 6 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**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 vertex *v* ∈ *V*(*G*), *N*(*v*) = {*x* ∈ *V*(*G*) : *xv* ∈ *E*(*G*)} and *N*[*v*] = *N*(*v*) ∪ {*v*}. A vertex *v* ∈ *V*(*G*) is called a *leaf vertex* if |*N*(*v*)| = 1. Given a set *D* ⊆ *V*(*G*), *N*(*D*) = ∪*v*∈*DN*(*v*), *N*[*D*] = *N*(*D*) ∪ *D* and *∂*(*D*) = *N*(*D*) \ *D*. Moreover, given a set *D* ⊆ *V*(*G*) and a vertex *v* ∈ *D*, *epn*(*v*, *D*) = {*u* ∈ *V*(*G*) \ *D* : *N*(*u*) ∩ *D* = {*v*}}. Also, and as is commonly defined, *G* − *D* denotes the graph obtained from *G* such that *V*(*G* − *D*) = *V*(*G*) \ *D* and *E*(*G* − *D*) = *E*(*G*) \ {*uv* ∈ *E*(*G*) : *u* ∈ *D* or *v* ∈ *D*}. Moreover, the subgraph of *G* induced by *D* ⊆ *V*(*G*) will be denoted by *G*[*D*].

We say that *G* is *F*-*free* if it contains no copy of *F* as an induced subgraph. A set *D* ⊆ *V*(*G*) is a 2-*packing* if *N*[*x*] ∩ *N*[*y*] = ∅ for every pair *x*, *y* ∈ *D*. The 2-*packing number* of *G*, denoted by *ρ*(*G*), is defined as max{|*D*| : *D* is a 2-packing of *G*}. A 2-packing of cardinality *ρ*(*G*) is called a *ρ*(*G*)-*set*. We will assume an analogous correspondence when referring to the optimal sets or functions derived from other parameters used in the article.

Let *f* : *V*(*G*) → {0, 1, 2} be a function on *G*. Observe that *f* generates three sets *V*0, *V*<sup>1</sup> and *V*2, where *Vi* = {*v* ∈ *V*(*G*): *f*(*v*) = *i*} for *i* ∈ {0, 1, 2}. In this sense, we will write *<sup>f</sup>*(*V*0, *<sup>V</sup>*1, *<sup>V</sup>*2) to refer to the function *<sup>f</sup>* . Given a set *<sup>D</sup>* ⊆ *<sup>V</sup>*(*G*), *<sup>f</sup>*(*D*) = <sup>∑</sup>*v*∈*<sup>D</sup> <sup>f</sup>*(*v*). The *weight* of *f* is defined as *ω*(*f*) = *f*(*V*(*G*)) = |*V*1| + 2|*V*2|. We shall also use the following notations: *V*1,2 = {*v* ∈ *V*<sup>1</sup> : *N*(*v*) ∩ *V*<sup>2</sup> = ∅}, *V*1,0 = {*v* ∈ *V*<sup>1</sup> : *N*(*v*) ⊆ *V*0} and *V*1,1 = *V*<sup>1</sup> \ (*V*1,2 ∪ *V*1,0). A function *f*(*V*0, *V*1, *V*2) on *G* is a *dominating function* if *N*(*v*) ∩ (*V*<sup>1</sup> ∪ *V*2) = ∅ for every vertex *v* ∈ *V*0. Moreover, *f* is a *total dominating function* (TDF) if *N*(*v*) ∩ (*V*<sup>1</sup> ∪ *V*2) = ∅ for every vertex *v* ∈ *V*(*G*). Next, we highlight some particular cases of known domination parameters, which we define here in terms of (total) dominating functions.


As consequence of the above definitions and the well-known inequalities *ρ*(*G*) ≤ *γ*(*G*) (see [1]), *γt*(*G*) ≤ *γR*(*G*) (see [21]) and *γtR*(*G*) ≤ *γR*(*G*) + *γ*(*G*) (see [14]), we establish an inequality chain involving the previous parameters.

**Theorem 1.** *If G is a graph with no isolated vertex, then*

*ρ*(*G*) ≤ *γ*(*G*) ≤ *γt*(*G*) ≤ *γR*(*G*) ≤ *γqtR*(*G*) ≤ *γtR*(*G*) ≤ *γR*(*G*) + *γ*(*G*).

For instance, for the graphs *G*<sup>1</sup> and *G*<sup>2</sup> given in Figure 1 we deduce the next inequality chains. In that regard, the labels of (gray and black) coloured vertices describe the positive weights of a *γqtR*(*Gi*)-function, for *i* ∈ {1, 2}.


**Figure 1.** The labels of (gray and black) coloured vertices describe the positive weights of a *γqtR*(*Gi*) function, for *i* ∈ {1, 2}.

As mentioned before, the goal of this work is continue the study of the quasi-total Roman domination number of graphs. In that regard, the paper is organized as follows. First, we obtain new, tight bounds for this parameter. Such bounds can also be seen as relationships between this novel parameter and several other classical domination parameters such as the (total) domination and (total) Roman domination numbers. Finally, and as a consequence of this previous study, we derive new results on the total Roman domination number of a graph.

#### **2. Bounds and Relationships with Other Parameters**

Let *G* be a disconnected graph and let *G*1, ... , *Gr* (*r* ≥ 2) be the components of *G*. Observe that any QTRDF *f*(*V*0, *V*1, *V*2) on *G* satisfies that *f* restricted to *V*(*Gj*) is a QTRDF on *Gj*, for every *j* ∈ {1, ... ,*r*}. Therefore, the following result is obtained for the case of disconnected graphs.

**Remark 1** ([6])**.** *Let G*1,..., *Gr (r* ≥ 2*) be the components of a disconnected graph G. Then*

$$\gamma\_{qtR}(G) = \sum\_{i=1}^{r} \gamma\_{qtR}(G\_i).$$

As a consequence of the above remark, throughout this paper, we only consider nontrivial connected graphs. Next, we give two useful lemmas, which provide some tools to deduce some of the results.

**Lemma 1.** *Let G be a nontrivial connected graph. If f*(*V*0, *V*1, *V*2) *is a γqtR*(*G*)*-function, then the following statements hold.*

(i) *f* (*V* <sup>0</sup> = *V*0, *V* <sup>1</sup> = *V*<sup>1</sup> \ *V*1,0, *V* <sup>2</sup> = *V*2) *is a γtR*(*G* − *V*1,0)*-function.* (ii) *epn*(*v*, *V*2) ∩ *V*<sup>0</sup> = ∅*, for every v* ∈ *V*2*.* (iii) *If γqtR*(*G*) = *γR*(*G*)*, then V*1,2 = ∅*.*

**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γqtR*(*G*)-function. First, we proceed to prove (i). Notice that *G* − *V*1,0 has no isolated vertex. Hence, the function *f* (*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>), defined by *V* <sup>0</sup> = *V*0, *V* <sup>1</sup> = *V*<sup>1</sup> \ *V*1,0 and *V* <sup>2</sup> = *V*2, is a TRDF on *G* − *V*1,0. Hence, *γtR*(*G* − *V*1,0) ≤ *ω*(*f* ). Now, if *γtR*(*G* − *V*1,0) < *ω*(*f* ), then from any *γtR*(*G* − *V*1,0)-function and the set *V*1,0, we can construct a QTRDF on *G* of weight less than *ω*(*f*) = *γqtR*(*G*), which is a contradiction. Therefore, the function *f* is a *γtR*(*G* − *V*1,0)-function, as required.

Now, we proceed to prove (ii). Let *v* ∈ *V*2. Obviously, *N*(*v*) ∩ *V*<sup>0</sup> = ∅. If *epn*(*v*, *V*2) ∩ *V*<sup>0</sup> = ∅, then the function *f* (*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>), defined by *V* <sup>1</sup> = *V*<sup>1</sup> ∪ {*v*}, *V* <sup>2</sup> = *V*<sup>2</sup> \ {*v*} and *V* <sup>0</sup> = *V*0, is a QTRDF on *G* of weight *ω*(*f* ) < *ω*(*f*) = *γqtR*(*G*), which is a contradiction. Therefore, *epn*(*v*, *V*2) ∩ *V*<sup>0</sup> = ∅, which completes the proof of (ii).

Finally, we proceed to prove (iii). Assume that *γqtR*(*G*) = *γR*(*G*). First, suppose that *V*1,2 = ∅. It is easy to see that the function *f* (*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>), defined by *V* <sup>1</sup> = *V*<sup>1</sup> \ *V*1,2, *V* <sup>2</sup> = *V*<sup>2</sup> and *V* <sup>0</sup> = *V*<sup>0</sup> ∪ *V*1,2, is a Roman-dominating function on *G*. Hence, *γR*(*G*) ≤ *ω*(*f* ) <

*ω*(*f*) = *γqtR*(*G*), which is a contradiction. Therefore, *V*1,2 = ∅, which completes the proof of (iii).

**Lemma 2.** *Let G be a nontrivial connected graph. If f*(*V*0, *V*1, *V*2) *is a γqtR*(*G*)*-function such that* |*V*1| *is minimum, then one of the following conditions holds.*


**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γqtR*(*G*)-function, such that |*V*1| is minimal. Assume that *V*1,0 = ∅. It is clear by definition that *V*1,0 is an independent set. Now, suppose that *V*1,0 is not a 2-packing of *G*. Therefore, two vertices *u*, *v* ∈ *V*1,0 exist at distance two. Let *w* ∈ *N*(*u*) ∩ *N*(*v*). Notice that *w* ∈ *V*<sup>0</sup> and *N*(*w*) ∩ *V*<sup>2</sup> = ∅. With these conditions in mind, observe that the function *f* (*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>), defined by *V* <sup>1</sup> = *V*<sup>1</sup> \ {*u*, *v*}, *V* <sup>2</sup> = *V*<sup>2</sup> ∪ {*w*} and *V* <sup>0</sup> = *V*(*G*) \ (*V* <sup>1</sup> ∪ *V* <sup>2</sup>), is a QTRDF on *G* of weight *ω*(*f* ) = *ω*(*f*) and |*V* <sup>1</sup>| < |*V*1|, which is a contradiction. Therefore, *V*1,0 is a 2-packing of *G*, which completes the proof.

We continue with one of the main results of this paper.

**Theorem 2.** *If G is a nontrivial connected graph, then at least one of the following statements holds.*


**Proof.** Let *f*(*V*0, *V*1, *V*2) be a *γqtR*(*G*)-function such that |*V*1| is minimum. If *V*1,0 = ∅, then by Lemma 1-(i) we deduce that *f* is also a *γtR*(*G*)-function, which implies that *γqtR*(*G*) = *γtR*(*G*). Hence, from now on, we assume that *V*1,0 = ∅. By Lemma 2, it follows that *V*1,0 is a 2-packing of *G*. Moreover, by Lemma 1-(i) we have the function *f* (*V* <sup>0</sup> = *V*0, *V* <sup>1</sup> = *V*<sup>1</sup> \ *V*1,0, *V* <sup>2</sup> = *V*2) is a *γtR*(*G* − *V*1,0)-function. Therefore, *γqtR*(*G*) = *γtR*(*G* − *V*1,0) + |*V*1,0| ≥ min{*γtR*(*G* − *S*) + |*S*| : *S* is a 2-packing of *G*}. We only need to prove that *γqtR*(*G*) ≤ min{*γtR*(*G* − *S*) + |*S*| : *S* is a 2-packing of *G*}. In such a sense, let *S* be a 2-packing of *G* for which *γtR*(*G* − *S*) + |*S*| is minimum, and let *g* (*W* <sup>0</sup>, *W* <sup>1</sup>, *W* <sup>2</sup>) be a *γtR*(*G* − *S*)-function. Observe that the function *g*(*W*0, *W*1, *W*2), defined by *W*<sup>0</sup> = *W* 0, *W*<sup>1</sup> = *W* <sup>1</sup> ∪ *S* and *W*<sup>2</sup> = *W* <sup>2</sup>, is a QTRDF on *G*. Therefore, *γqtR*(*G*) ≤ *ω*(*g*) = min{*γtR*(*G* − *S*) + |*S*| : *S* is a 2-packing of *G*}, which completes the proof.

The next proposition is a direct consequence of Theorem 2.

**Proposition 1.** *If G is a nontrivial connected graph, then*

$$
\gamma\_{\mathfrak{q}tR}(G) \ge \gamma\_{tR}(G) - \rho(G).
$$

**Proof.** If *γqtR*(*G*) = *γtR*(*G*), then the inequality holds. Assume that *γqtR*(*G*) < *γtR*(*G*). By Theorem 2 there exists a 2-packing *S* of *G* such that *γqtR*(*G*) = *γtR*(*G* − *S*) + |*S*|. Let *f* (*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>) be a *γtR*(*G* − *S*)-function and let *S* ⊆ *N*(*S*) be a set of cardinality |*S*| such that *N*(*x*) ∩ *S* = ∅ for every vertex *x* ∈ *S*. Observe that the function *f*(*V*0, *V*1, *V*2), defined by *V*<sup>2</sup> = *V* <sup>2</sup>, *V*<sup>1</sup> = *V* <sup>1</sup> ∪ *S* ∪ (*S* \ *V* <sup>2</sup>) and *V*<sup>0</sup> = *V*(*G*) \ (*V*<sup>1</sup> ∪ *V*2), is a TRDF on *G*. Therefore, *γtR*(*G*) ≤ *ω*(*f*) ≤ *ω*(*f* ) + |*S*| + |*S* | = *γtR*(*G* − *S*) + 2|*S*| = *γqtR*(*G*) + |*S*| ≤ *γqtR*(*G*) + *ρ*(*G*), which completes the proof.

The bound above is tight. For instance, it is achieved for the graph *G* given in the Figure 2. Notice that this figure describes the positive weights of a *γqtR*(*G*)-function. In addition, it is easy to see that *ρ*(*G*) = 2 and *γtR*(*G*) = 8. Hence, *γqtR*(*G*) = 6 = *γtR*(*G*) − *ρ*(*G*), as required.

**Figure 2.** The labels of (gray and black) coloured vertices describe the positive weights of a *γqtR*(*G*)-function.

It is well-known that *γtR*(*G*) ≥ 2*γ*(*G*) ≥ *γR*(*G*) for any graph *G* with no isolated vertex (see [3,15]). From this inequality chain, we deduce the following result.

**Theorem 3.** *For any nontrivial connected graph G,*

$$
\mathfrak{a}\gamma(\mathcal{G}) - \rho(\mathcal{G}) \le \gamma\_{qtR}(\mathcal{G}) \le \mathfrak{z}\gamma(\mathcal{G}).
$$

**Proof.** By combining the bound *γtR*(*G*) ≥ 2*γ*(*G*) and the bound given in Proposition 1, we deduce that *γqtR*(*G*) ≥ 2*γ*(*G*) − *ρ*(*G*).

Now, from the bound *γR*(*G*) ≤ 2*γ*(*G*) and the inequality *γqtR*(*G*) ≤ *γR*(*G*) + *γ*(*G*) given in Theorem 1 we obtain *γqtR*(*G*) ≤ *γR*(*G*) + *γ*(*G*) ≤ 3*γ*(*G*), as desired.

The lower bounds given in the two previous results are tight. We will show later that, as a consequence of Lemma 3, the graphs *Ga*,0 ∈ G satisfy the equality established in Proposition 1, while the graph *G*2,0 satisfies the equality given in Theorem 3.

With respect to the equality in the bound *γqtR*(*G*) ≤ 3*γ*(*G*) above, we can see that this bound is tight. For instance, it is achieved for the graph *G* given in the Figure 3. Notice that this figure describes the positive weights of a *γqtR*(*G*)-function, and as a consequence, we deduce that *γqtR*(*G*) = 6 = 3*γ*(*G*), as required.

**Figure 3.** The labels of (gray and black) coloured vertices describe the positive weights of a *γqtR*(*G*)-function.

In addition, we can deduce the following connection. To this end, we need to say that a graph *G* is called a *Roman graph* if *γR*(*G*) = 2*γ*(*G*).

**Proposition 2.** *If G is a graph such that γqtR*(*G*) = 3*γ*(*G*)*, then G is a Roman graph.*

**Proof.** From the proof of Theorem 3, we have that 3*γ*(*G*) = *γqtR*(*G*) ≤ *γR*(*G*) + *γ*(*G*) ≤ 3*γ*(*G*). Thus, we have equalities in the inequality chain above. In particular, *γR*(*G*) = 2*γ*(*G*), which completes the proof.

Notice that the opposed to the proposition above is not necessarily true. For instance, the graph *G*<sup>2</sup> given in Figure 1 is a Roman graph, but it does not satisfy the equality *γqtR*(*G*2) = 3*γ*(*G*2).

The following result gives a lower bound for the quasi-total Roman domination number and characterizes the class of connected graphs for which *γqtR*(*G*) ∈ {*γ*(*G*) + 1, *γ*(*G*) + 2}.

**Theorem 4.** *For any nontrivial connected graph G of order n,*

$$
\gamma\_{qtR}(G) \ge \gamma(G) + 1.
$$

*Furthermore,*

	- (a) *G* ∼= *P*<sup>2</sup> *has a vertex of degree n* − *γ*(*G*)*.*
	- (b) *G has two adjacent vertices u*, *v such that* |*∂*({*u*, *v*})| = *n* − *γ*(*G*)*.*

**Proof.** If *G* ∼= *P*2, then it is clear that *γqtR*(*G*) = *γ*(*G*) + 1. From now on, assume that *G* ∼= *P*2. Let *f*(*V*0, *V*1, *V*2) be a *γqtR*(*G*)-function, such that |*V*1| is minimum. Note that (*V*<sup>1</sup> \ *V*1,2) ∪ *V*<sup>2</sup> is a dominating set of *G*, and |*V*2| ≥ 1. Hence, *γqtR*(*G*) = 2|*V*2| + |*V*1| ≥ (|*V*2| + |*V*<sup>1</sup> \ *V*1,2|) + |*V*2| ≥ *γ*(*G*) + 1, and the lower bound follows.

Now, suppose that *γqtR*(*G*) = *γ*(*G*) + 1. So, we have equalities in the inequality chain above. In particular, *V*1,2 = ∅ and |*V*2| = 1, which is a contradiction. Therefore, if *G* ∼= *P*2, then *γqtR*(*G*) ≥ *γ*(*G*) + 2, and, as a consequence, (i) follows.

We next proceed to prove (ii). First, suppose that *γqtR*(*G*) = *γ*(*G*) + 2. Notice that,

$$
\gamma(G) + 2 = \omega(f) \ge \left( |V\_2| + |V\_1 \backslash V\_{1,2}| \right) + |V\_2| \ge \gamma(G) + |V\_2|.
$$

This implies that |*V*2|∈{1, 2}, and, in such a case, we consider the following two cases. Case 1. |*V*2| = 1. In this case, we have that |*V*1| = *γ*(*G*). Let *V*<sup>2</sup> = {*v*}. Now, as |*N*(*v*) ∩ *V*1| = 1 and *V*<sup>0</sup> ⊆ *N*(*v*), we deduce that |*N*(*v*)| = |*V*0| + 1 = (*n* − |*V*1|−|*V*2|) + 1 = *n* − *γ*(*G*), which implies that condition (a) follows.

Case 2. |*V*2| = 2. Let *V*<sup>2</sup> = {*u*, *v*}. In this case we have that |*V*1| = *γ*(*G*) − 2, and we have equalities in the inequality chain above. As a consequence, *V*1,2 = ∅, which implies that *u* and *v* are adjacent vertices. Hence, *∂*({*u*, *v*}) = *V*<sup>0</sup> and, therefore, |*∂*({*u*, *v*})| = |*V*0| = *n* − |*V*1|−|*V*2| = *n* − *γ*(*G*). Therefore, condition (b) follows.

On the other hand, suppose that one of the conditions (a) and (b) holds. In such a sense, we consider the next two cases. Recall that *γqtR*(*G*) ≥ *γ*(*G*) + 2 since *G* ∼= *P*2.

Case 1. Suppose that (a) holds. Let *v* ∈ *V*(*G*) such that |*N*(*v*)| = *n* − *γ*(*G*) and *w* ∈ *N*(*v*). Notice that the function *f* (*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>), defined by *V* <sup>2</sup> = {*v*}, *V* <sup>0</sup> = *N*(*v*) \ {*w*} and *V* <sup>1</sup> = *V*(*G*) \ (*V* <sup>0</sup> ∪ *V* <sup>2</sup>), is a QTRDF on *G*. Hence, *γqtR*(*G*) ≤ *ω*(*f* ) = 2|*V* <sup>2</sup>| + |*V* <sup>1</sup>| = 2 + *γ*(*G*), which implies that *γqtR*(*G*) = *γ*(*G*) + 2, as required.

Case 2. Suppose that (b) holds. Let *u*, *v* be two adjacent vertices such that |*∂*({*u*, *v*})| = *n* − *γ*(*G*). Observe that the function *f* (*V* <sup>0</sup> , *V* <sup>1</sup> , *V* <sup>2</sup> ), defined by *V* <sup>2</sup> = {*u*, *v*}, *V* <sup>0</sup> = *∂*({*u*, *v*}) and *V* <sup>1</sup> = *V*(*G*) \ (*V* <sup>0</sup> ∪ *V* <sup>2</sup> ), is a QTRDF on *G*. Hence, *γqtR*(*G*) ≤ *ω*(*f* ) = 2|*V* <sup>2</sup> | + |*V* <sup>1</sup> | = 4 + (*γ*(*G*) − 2) = *γ*(*G*) + 2, which implies that *γqtR*(*G*) = *γ*(*G*) + 2, as required.

Therefore, the proof is complete.

Cabrera Martínez et al. [6] in 2019, established that *γqtR*(*G*) ≤ *n* − *ρ*(*G*)(*δ*(*G*) − 2) for any nontrivial graph *G* of order *n* and minimum degree *δ*(*G*). The following bounds for the total Roman domination number and the domination number, respectively, are direct consequences of the previous inequality, Proposition 1 and Theorem 3.

**Theorem 5.** *The following statements hold for any nontrivial connected graph G of order n and δ*(*G*) ≥ 4*.*


From Proposition 1 and Theorem 1, we obtain the following useful inequality chain.

$$
\gamma\_{tR}(G) - \rho(G) \le \gamma\_{qtR}(G) \le \gamma\_{tR}(G). \tag{1}
$$

An interesting question that arises from the inequality chain above is the following. Can the differences *γqtR*(*G*) − (*γtR*(*G*) − *ρ*(*G*)) and *γtR*(*G*) − *γqtR*(*G*) be as large as possible? Next, we provide an affirmative answer to the previous question. For this purpose, we

need to introduce the following family of graphs. Given two integers *a*, *b* ≥ 0 (*a* + *b* ≥ 2), a graph *Ga*,*<sup>b</sup>* ∈ G is defined as follows.


The Figure 4 shows the graph *G*2,3 by taking *G* ∼= *P*5. We next give exact formulas for the total Roman domination number, the quasi-total Roman domination number and the packing number of the graphs of the family G. These results are almost straightforward to deduce and, according to this fact, the proofs are left to the reader.

**Figure 4.** The graph *G*2,3 by taking *G* as the path graph *P*5. The labels of (gray and black) coloured vertices describe the positive weights of a *γqtR*(*G*2,3)-function.

**Lemma 3.** *Let a*, *b* ≥ 0 *be two integers, such that a* + *b* ≥ 2*. If G is a connected graph such that* |*V*(*G*)| = *a* + *b, then the following equalities hold.*


According to the lemma above, for any integers *a*, *b* ≥ 0 (*a* + *b* ≥ 2), we obtain that any graph *Ga*,*<sup>b</sup>* ∈ G satisfies

$$
\gamma\_{qlR}(G\_{a,b}) - (\gamma\_{lR}(G\_{a,b}) - \rho(G\_{a,b})) = b \quad \text{and} \quad \gamma\_{lR}(G\_{a,b}) - \gamma\_{qlR}(G\_{a,b}) = a\_{\prime\prime}
$$

which provides the answer to our previous question. In addition, and as a consequence of Lemma 3, we deduce that the lower and upper bounds given in Inequality chain (1) are tight. For instance, any graph *Ga*,0 ∈ G satisfies that *γqtR*(*Ga*,0) = *γtR*(*Ga*,0) − *ρ*(*Ga*,0), while any graph *G*0,*<sup>b</sup>* ∈ G satisfies that *γqtR*(*G*0,*b*) = *γtR*(*G*0,*b*).

It is well known that *ρ*(*G*) = 1 for every graph *G* with a diameter of, at most, two. In this sense, and as direct consequence of the Inequality chain (1), we have that *γqtR*(*G*) ∈ {*γtR*(*G*) − 1, *γtR*(*G*)} for every graph *G* with diameter of, at most, two. We next show some subclasses which satisfy the equality *γqtR*(*G*) = *γtR*(*G*). For this, we need to cite the following result.

**Theorem 6** ([6])**.** *The following statements hold for any nontrivial graph G.*


The join of two graphs *G*<sup>1</sup> and *G*2, denoted by *G*<sup>1</sup> + *G*2, is the graph obtained from *G*<sup>1</sup> and *G*<sup>2</sup> with vertex set *V*(*G*<sup>1</sup> + *G*2) = *V*(*G*1) ∪ *V*(*G*2) and edge set *E*(*G*<sup>1</sup> + *G*2) = *E*(*G*1) ∪ *E*(*G*2) ∪ {*uv* : *u* ∈ *V*(*G*1), *v* ∈ *V*(*G*2)}. Observe that *diam*(*G*<sup>1</sup> + *G*2) ≤ 2 by definition. The following result, which is a consequence of Theorem 6, shows that *γtR*(*G*<sup>1</sup> + *G*2) = *γqtR*(*G*<sup>1</sup> + *G*2).

**Theorem 7.** *For any nontrivial graphs G*<sup>1</sup> *and G*2*,*

$$\gamma\_{\eta lR}(G\_1 + G\_2) = \gamma\_{lR}(G\_1 + G\_2) = \begin{cases} \mathfrak{Z}\_{\prime} & \text{if } \min\{\gamma(G\_1), \gamma(G\_2)\} = 1;\\ 4, & \text{otherwise}. \end{cases}$$

We continue analysing other subclasses of graphs with a diameter of two. The following results consider the planar graphs with a diameter of two.

**Theorem 8** ([22])**.** *If G is a planar graph with diam*(*G*) = 2*, then the following statements hold.* (i) *γ*(*G*) ≤ 2 *or G* = *G*9*, where G*<sup>9</sup> *is the graph given in Figure 5.* (ii) *γt*(*G*) ≤ 3*.*

**Figure 5.** The planar graph *G*<sup>9</sup> with *diam*(*G*9) = 2 and *γt*(*G*9) = *γ*(*G*9) = 3.

**Theorem 9.** *For any planar graph G with diam*(*G*) = 2*,*

$$\gamma\_{qtR}(\mathcal{G}) = \gamma\_{tR}(\mathcal{G}) = \begin{cases} \mathfrak{Z}, & \text{if } \gamma(\mathcal{G}) = 1; \\ 4, & \text{if } \gamma(\mathcal{G}) = \gamma\_t(\mathcal{G}) = 2; \\ 5, & \text{if } \gamma\_t(\mathcal{G}) = \gamma(\mathcal{G}) + 1 = 3; \\ 6, & \text{if } \mathcal{G} = \mathcal{G}\_9. \end{cases}$$

**Proof.** If *G* = *G*9, then it is easy to check that *γqtR*(*G*) = *γtR*(*G*) = 6. From now on, let *G* = *G*<sup>9</sup> be a planar graph with *diam*(*G*) = 2. It is straightforward that *γqtR*(*G*) = *γtR*(*G*) = 3 if and only if *γ*(*G*) = 1. Hence, assume that *γ*(*G*) ≥ 2. By Theorem 8, it follows that *γ*(*G*) = 2 and *γt*(*G*) ∈ {2, 3}. Next, we analyse these two cases.

Case 1. *γt*(*G*) = 2. By Theorems 6 and 1 and the well-known bound *γtR*(*G*) ≤ 2*γt*(*G*) (see [15]) we obtain that 4 = *γqtR*(*G*) ≤ *γtR*(*G*) ≤ 2*γt*(*G*) = 4. Thus, *γqtR*(*G*) = *γtR*(*G*) = 4. Case 2. *γt*(*G*) = 3. As a consequence of the Theorem 6 we have that *γqtR*(*G*) ≥ 5. Let {*u*, *v*} be a *γ*(*G*)-set. Since *γt*(*G*) = 3 and *diam*(*G*) = 2, it follows that *u* and *v* are at distance two. Let *w* ∈ *N*(*u*) ∩ *N*(*v*). Notice that the function *f* , defined by *f*(*u*) = *f*(*v*) = 2, *f*(*w*) = 1 and *f*(*x*) = 0 for every *x* ∈ *V*(*G*) \ {*u*, *v*, *w*}, is a TRDF on *G*, which implies that *γtR*(*G*) ≤ *ω*(*f*) = 5. Hence, by the fact that *γqtR*(*G*) ≤ *γtR*(*G*) we deduce that *γqtR*(*G*) = *γtR*(*G*) = 5.

Therefore, the proof is complete.

However, for the case of non-planar graphs with a diameter of two, there are graphs that satisfy *γqtR*(*G*) = *γtR*(*G*) or *γqtR*(*G*) = *γtR*(*G*) − 1. For instance, for the graphs *G*<sup>1</sup> and *G*<sup>2</sup> given in Figure 1 we have that *γqtR*(*G*1) = 6 = *γtR*(*G*1) −1 and *γqtR*(*G*2) = 7 = *γtR*(*G*2). In connection with this fact, we pose the following open problem.

**Problem 1.** *Characterize the families of non-planar graphs G with diameter two for which γqtR*(*G*) = *γtR*(*G*) *or γqtR*(*G*) = *γtR*(*G*) − 1*.*

Notice that, as consequence of the Inequality chain (1), any new result for the total Roman domination number gives us a new result for the quasi-total Roman domination number and vice versa. In such a sense, we continue with two new bounds for the total Roman domination number. Before this, we need to introduce the following definition. A set *S* of vertices of a graph *G* is a *vertex cover* if every edge of *G* is incident with at least one vertex in *S*. The *vertex cover number* of *G*, denoted by *β*(*G*), is the minimum cardinality among all vertex covers of *G*.

**Theorem 10.** *For any K*1,3*-free graph G with δ*(*G*) ≥ 3*,*

$$
\gamma\_{tR}(G) \le \beta(G) + \gamma(G).
$$

**Proof.** Let *D* be a *γ*(*G*)-set and *S* a *β*(*G*)-set. Let *f*(*V*0, *V*1, *V*2) be a function defined by *V*<sup>0</sup> = *V*(*G*) \ (*D* ∪ *S*), *V*<sup>1</sup> = (*D* ∪ *S*) \ (*D* ∩ *S*) and *V*<sup>2</sup> = *D* ∩ *S*. Now, we proceed to prove that *f* is a TRDF on *G*. We first note that *S* is a total dominating set because *G* is *K*1,3-free. Hence, *V*<sup>1</sup> ∪ *V*<sup>2</sup> = *D* ∪ *S* is a total dominating set of *G*. Let *v* ∈ *V*<sup>0</sup> = *V*(*G*) \ (*D* ∪ *S*). So, *N*(*v*) ⊆ *S* and *N*(*v*) ∩ *D* = ∅. Hence *N*(*v*) ∩ *D* ∩ *S* = ∅, i.e., *N*(*v*) ∩ *V*<sup>2</sup> = ∅. Therefore, *f* is a TRDF on *G*, as desired. Thus, *γtR*(*G*) ≤ *ω*(*f*) ≤ |(*D* ∪ *S*) \ (*D* ∩ *S*)| + 2|*D* ∩ *S*| = *β*(*G*) + *γ*(*G*), which completes the proof.

**Lemma 4** ([15])**.** *If G is a graph with no isolated vertex, then there exists a γtR*(*G*)*-function f*(*V*0, *V*1, *V*2) *such that either V*<sup>2</sup> *is a dominating set of G, or the set S of vertices not dominated by V*<sup>2</sup> *satisfies G*[*S*] = *kK*<sup>2</sup> *for some k* ≥ 1*, where S* ⊆ *V*<sup>1</sup> *and ∂*(*S*) ⊆ *V*0*.*

**Theorem 11.** *If G is a* {*K*1,3, *K*1,3 + *e*}*-free graph such that δ*(*G*) ≥ 3*, then there exists a γtR*(*G*) *function f*(*V*0, *V*1, *V*2) *such that V*<sup>2</sup> *is a dominating set of G, and, as a consequence,*

$$
\gamma\_{tR}(G) \ge \gamma\_t(G) + \gamma(G).
$$

**Proof.** Suppose that there is no *γtR*(*G*)-function *f*(*V* <sup>0</sup>, *V* <sup>1</sup>, *V* <sup>2</sup>) such that *V* <sup>2</sup> is a dominating set of *G*. By Lemma 4, there exists a *γtR*(*G*)-function *f*(*V*0, *V*1, *V*2) such that *V*1,1 satisfies that *G*[*V*1,1] = *kK*<sup>2</sup> for some *k* ≥ 1 and *∂*(*V*1,1) ⊆ *V*0. We can assume that |*V*1| is minimum among all *γtR*(*G*)-functions because it is a requirement for the existence of the function *f* (see the proof of Lemma 4). Let *u*, *v* ∈ *V*1,1 be two adjacent vertices. Hence, *∂*({*u*, *v*}) ⊆ *V*0. Since *δ*(*G*) ≥ 3, there are two vertices *w*1, *w*<sup>2</sup> ∈ *N*(*v*) ∩ *V*0, and as *G* is a {*K*1,3, *K*1,3 + *e*}-free graph, we deduce that at least one of these vertices is also adjacent to *u*. Hence, and without loss of generality, assume that {*u*, *v*} ⊆ *N*(*w*1). Observe that the function *g*(*W*0, *W*1, *W*2), defined by *W*<sup>2</sup> = *V*<sup>2</sup> ∪ {*w*1}, *W*<sup>1</sup> = *V*<sup>1</sup> \ {*u*, *v*} and *W*<sup>0</sup> = *V*(*G*) \ (*W*<sup>1</sup> ∪ *W*2), is a TRDF on *G* of weight *ω*(*g*) = *ω*(*f*) and |*W*1| < |*V*1|, which is a contradiction. Therefore, there exists a *γtR*(*G*)-function *f*(*V*0, *V*1, *V*2) such that *V*<sup>2</sup> is a dominating set of *G*. Since *V*<sup>1</sup> ∪ *V*<sup>2</sup> is a total dominating set of *G*, we deduce that *γt*(*G*) + *γ*(*G*) ≤ |*V*<sup>1</sup> ∪ *V*2| + |*V*2| = 2|*V*2| + |*V*1| = *γtR*(*G*), which completes the proof.

Observe that, if *G* is a {*K*1,3, *K*1,3 + *e*}-free graph of minimum degree at least three with *β*(*G*) = *γt*(*G*), then the bounds given in the two previous theorems are achieved. Moreover, let *G* be a (*n* − 2)-regular graph obtained from the complete graph *Kn* (*n* even) by deleting the edges of a perfect matching. Notice that *G* is {*K*1,3, *K*1,3 + *e*}-free and satisfies that *γtR*(*G*) = 4 = *γt*(*G*) + *γ*(*G*).

**Theorem 12.** *If G is a connected* {*K*1,3, *K*1,3 + *e*}*-free graph such that δ*(*G*) ≥ 3*, then the following statements hold.*

*(i) γt*(*G*) + *γ*(*G*) − *ρ*(*G*) ≤ *γqtR*(*G*) ≤ *β*(*G*) + *γ*(*G*). *(ii) If γtR*(*G*) = *γR*(*G*)*, then γqtR*(*G*) = 2*γt*(*G*).

**Proof.** Statement (i) is a direct consequence of combining Inequality chain (1) and Theorems 10 and 11. Finally, we proceed to prove (ii). By Theorem 11 there exists a *γtR*(*G*) function *f*(*V*0, *V*1, *V*2) such that *V*<sup>2</sup> is a dominating set of *G*. Hence, *V*1,1 = ∅. Moreover, as *γtR*(*G*) = *γR*(*G*), we deduce that *f* is also a *γqtR*(*G*)-function and Lemma 1 (iii)-(a) leads

to *V*1,2 = ∅. Therefore *V*<sup>1</sup> = ∅, which implies that *V*<sup>2</sup> is a total dominating set of *G*. Hence, 2*γt*(*G*) ≤ 2|*V*2| = *γqtR* = *γtR* ≤ 2*γt*(*G*). Therefore, *γqtR* = 2*γt*(*G*), as required.

#### **3. Conclusions and Open Problems**

This paper is a contribution to the graph domination theory. We have studied the quasitotal Roman domination in graphs. For instance, we have shown the close relationship that exists between this novel parameter and other invariants, such as (total) domination number, (total) Roman domination number and 2-packing number.

We conclude by proposing some open problems.

	- **–** *γqtR*(*G*) = *γtR*(*G*).
	- **–** *γqtR*(*G*) = *γtR*(*G*) − *ρ*(*G*).
	- **–** *γqtR*(*G*) = 3*γ*(*G*).

**Author Contributions:** Investigation, A.C.M., J.C.H.-G. and J.M.S. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Agencia Estatal de Investigación grant number PID2019- 106433GB-I00/AEI/10.13039/501100011033, Spain.

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

**Informed Consent Statement:** Not applicable.

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

#### **References**

