Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Abstract

For a simple graph $G=(V,E)$ with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function $f:V\left(G\right)\to \{0,1,2,3\}$ having the property that (i) ${\sum }_{w\in N\left(v\right)}f\left(w\right)\ge 3$ if $f\left(v\right)=0$; (ii) ${\sum }_{w\in N\left(v\right)}f\left(w\right)\ge 2$ if $f\left(v\right)=1$; and (iii) every vertex v with $f\left(v\right)\ne 0$ has a neighbor u with $f\left(u\right)\ne 0$ for every vertex $v\in V\left(G\right)$. The weight of a TR3DF f is the sum $f\left(V\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by ${\gamma }_{t\{R3\}}\left(G\right)$. In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of ${\gamma }_{t\{R3\}}$ for trees.

1. Introduction

Let $G=(V,E)$ be a graph with vertex set $V=V\left(G\right)$ and edge set $E=E\left(G\right)$. For every vertex $v\in V$, the open neighborhood $N_G\left(v\right)=N\left(v\right)=\{u\in V\left(G\right):uv\in E\left(G\right)\}$ and the closed neighborhood $N_G\left[v\right]=N\left[v\right]=N\left(v\right)\cup \{v\}$. We denote the degree of v by $d_G\left(v\right)=d\left(v\right)=|N_G\left(v\right)|$. A vertex of degree one is called a leaf and its neighbor is a support vertex, and a support vertex is called a strong support if it is adjacent to at least two leaves. Let $S_n$ be a star with order n. A tree T is an acyclic connected graph. $G=(G_1\cup G_2)$ is a union graph G such that $V\left(G\right)=V\left(G_1\right)\cup V\left(G_2\right)$ and $E\left(G\right)=E\left(G_1\right)\cup E\left(G_2\right)$.

Given a graph G and a positive integer k, assume that $f:V\left(G\right)\to \{0,1,2,...,k\}$ is a function, and suppose that $(V_0,V_1,..,V_k)$ is the ordered partition of V introduced by f, where $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for $i\in \{0,1,...,k\}$. Then we can write $f=(V_0,V_1,..,V_k)$ and ${\omega }_f\left(V\left(G\right)\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ is the weight of a function f of G.

A subset S of a vertex set $V\left(G\right)$ is a dominating set of G if for every vertex $v\in V\left(G\right)\backslash S$, there exists a vertex $w\in S$ such that $wv$ is an edge of G. The domination number of G denoted by $\gamma \left(G\right)$ is the smallest cardinality of a dominating set S of G [1]. A function $f:V\left(G\right)\to \{0,1\}$ is called a dominating function(DF) on G if every vertex u with $f\left(u\right)=0$ has a vertex $v\in N\left(u\right)$ such that $f\left(v\right)=1$ [2]. The dominating set problem(DSP) is to find the domination number of G, which has been deeply and widely studied in recent years [3,4,5,6,7].

A subset S of a vertex set $V\left(G\right)$ is a total dominating set of G if ${\bigcup }_{v\in S}N\left(v\right)=V\left(G\right)$. The total domination number of G denoted by ${\gamma }_t\left(G\right)$ is the smallest cardinality of a total dominating set S of G [8]. The literature on the subject of total domination in graphs has been surveyed and provided in detail in a recent book [9]. Moreover, Michael A. Henning et al. presented a survey of selected recent results on total domination in graphs [10].

The mathematical concept of Roman domination is originally defined and discussed by Stewart et al. [11] and ReVelle et al. [12]. A Roman dominating function(RDF) on graph G is a function $f:V\left(G\right)\to \{0,1,2\}$ such that every vertex $v\in V\left(G\right)$ for which $f\left(u\right)=0$ is adjacent to at least one vertex u with $f\left(u\right)=2$ [13]. The Roman domination number of G is the minimum weight overall $RDF$s, denoted by ${\gamma }_R\left(G\right)$ [14]. On the basis of Roman domination, signed Roman domination [15], double Roman domination [16] and total Roman domination [17] have been proposed recently.

The total Roman dominating function(TRDF) on G is an RDF f on G with an additional property that every vertex $v\in V\left(G\right)$ with $f \left(v\right)\ne 0$ has a neighbor u with $f \left(u\right)\ne 0$. Let ${\gamma }_{tR}\left(G\right)$ denote the minimum weight of all TRDFs on G. A TRDF on G with weight ${\gamma }_{tR}\left(G\right)$ is called a ${\gamma }_{tR}\left(G\right)$-function. The conception of TRDF was first defined by Hossein Ahangar et al. [18]. In addition, Nicolás Campanelli et al. studied the total Roman domination number of the lexicographic product of graphs [17] and Chloe Lampman et al. presented some basic results of Edge-Critical Graphs [19].

The Roman $\{2\}$-dominating function (also named Italian domination) f [20] introduced by Chellali et al. which is defined as follows: $f:V\left(G\right)\to \{0,1,2\}$ has the property that ${\sum }_{u\in N\left(v\right)}f\left(u\right)\ge 2$ for $f\left(v\right)=0$ [21]. Chellali et al. proved that the Roman $\{2\}$-domination problem is NP-complete for bipartite graphs [21]. Hangdi Chen showed that the Roman $\{2\}$-domination problem is NP-complete for split graphs, and gave a linear-time algorithm for finding the minimum weight of Roman $\{2\}$-dominating function in block graphs [22]. As a generalization of Roman domination, Michael A. Henning et al. studied the relationship between Roman $\{2\}$-domination and dominating set parameters in trees [20].

A Roman $\{3\}$-dominating function(R{3}DF) f defined by Mojdeh et al. [23], which is defined as follows: $f:V\left(G\right)\to \{0,1,2,3\}$ has the property that for every vertex $v\in V\left(G\right)$ with $f\left(v\right)\in \{0,1\}$ and ${\sum }_{u\in N\left(v\right)}f\left(u\right)\ge 3$. Mojdeh et al. presented an upper bound on the Roman $\{3\}$-domination number of a connected graph G, characterized the graphs attaining upper bound and showed that the Roman $\{3\}$-domination problem is NP-complete, even restricted to bipartite graphs [23].

The total Roman $\{3\}$-domination [24] was studied recently. The total Roman $\{3\}$-dominating function(TR3DF) on a graph G is an R{3}DF on G with the additional property that every vertex $v\in V\left(G\right)$ with $f\left(v\right)\ne 0$ has a neighbor w with $f\left(w\right)\ne 0$. The minimum weight of a total Roman $\{3\}$-dominating function on G denoted by ${\gamma }_{t\{R3\}}\left(G\right)$ is named the total Roman $\{3\}$-domination number of G. A ${\gamma }_{t\{R3\}}\left(G\right)$-function is a total Roman $\{3\}$-dominating function on G with weight ${\gamma }_{t\{R3\}}\left(G\right)$. Doost Ali Mojdeh et al. showed the relationship among total Roman $\{3\}$-domination, total domination, and total Roman$\{2\}$-domination parameters. They also presented an upper bound on the total Roman $\{3\}$-domination number of a connected graph G and characterized the graphs arriving this bound. Finally, they investigated that total Roman $\{3\}$-domination problem is NP-complete for bipartite graphs [24].

In this paper, we further investigate the complexity of total Roman $\{3\}$-domination in planar graphs and chordal bipartite graphs. Moreover, we give a linear-time algorithm to compute the ${\gamma }_{t\{R3\}}$ for trees which answer the problem that it is possible to construct a polynomial algorithm for computing the number of total Roman $\{3\}$-domination for trees [24].

2. Complexity

In this section, we study the complexity of total Roman {3}-domination of graph. We show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Consider the following decision problem.

• Total Roman {3}-Domination Problem TR3DP.
  Instance: Graph $G=(V,E)$, and a positive integer m.
  Question: Does G have a total Roman {3}-function with weight at most m?

Please note that the dominating set problem is NP-complete for planar graphs [25] and chordal bipartite graphs [26]. We show the NP-completeness results by reducing the well-known NP-complete problem, dominating set, to TR3D.

Let G be a graph on n vertices. Let $T_v$ be the tree with $V\left(T_v\right)=\{v,v_a,v_b,v_c,v_d,v_e,v_f,v_p,v_q\}$, $E\left(T_v\right)=\{v{v}_a,v_a{v}_c,v_c{v}_e,v_c{v}_f,v{v}_b,v_b{v}_d,v_d{v}_p,v_d{v}_q\}$, as depicted in Figure 1.

Let $G^{\prime }$ be the graph obtained by adding edges between $v^{\prime }\in T_{v^{\prime }}$ and $v^{″}\in T_{v^{″}}$ if $v^{\prime }{v}^{″}\in E\left(G\right)$ from the union of the trees $T_v$ for $v\in V\left(G\right)$. Please note that $|V\left(G^{\prime }\right)|=n\times |V\left(T_v\right)|=9n$ and $|E\left(G^{\prime }\right)|=|E\left(G\right)|+n\times |E\left(T_v\right)|=|E\left(G\right)|+8n$.

Lemma 1.

If G is a planar graph or chordal bipartite graph, so is $G^{\prime }$.

Lemma 2.

([24]) Let $S_n$ be a star with $n\ge 3$, then ${\gamma }_{t\{R3\}}\left(S_n\right)=4$.

Lemma 3.

Let g be a TR3DF of G. If v is a strong support vertex of G, then ${\omega }_g\left(N\left[v\right]\right)\ge 4$.

Proof of Lemma 3. 

Let $v_1,v_2,..,v_k$ be leaves of v with $k\ge 2$. Since $g\left(N\left[v_i\right]\right)\ge 3$ for $i\in \{1,2,..,k\}$, we have $g\left(v_i\right)\ge 3-g\left(v\right)$ for $i\in \{1,2,..,k\}$. Then ${\omega }_g\left(N\left[v\right]\right)=g\left(v\right)+{\sum }_{i\in \{1,2,...,k\}}g\left(v_i\right)\ge g\left(v\right)+g\left(v_1\right)+g\left(v_2\right)\ge 6-g\left(v\right)$. If $g\left(v\right)\le 2$, it is clear that ${\omega }_g\left(N\left[v\right]\right)\ge 4$. If $g\left(v\right)=3$, there exists a vertex $u\in N\left(v\right)$ with $g\left(u\right)\ne 0$. Then ${\omega }_g\left(N\left[v\right]\right)\ge 4$. □

Lemma 4.

If f is a $DF$ of G with ${\omega }_f\left(G\right)\le \ell$, then there exists a $TR3DF$ g of $G^{\prime }$ with ${\omega }_g\left(G^{\prime }\right)\le \ell +8n$.

Proof of Lemma 4. 

For each $v\in V\left(G\right)$, we define g as follows: $V\left(T_v\right)\to \{0,1,2,3\}$, $g\left(v_a\right)=g\left(v_b\right)=1$, $g\left(v_c\right)=g\left(v_d\right)=3$, $g\left(v\right)=f\left(v\right)$, $g\left(x\right)=0$ otherwise. It is clear that g is a $TR3DF$ of $G^{\prime }$. Therefore we have that ${\omega }_g\left(G^{\prime }\right)={\omega }_f\left(G\right)+8n\le \ell +8n$. □

Claim 1.

Let g be a $TR3DF$ of $G^{\prime }$, then ${\omega }_g\left(T_v^{\prime }\right)\ge 8$.

Proof of Claim 1. 

By Lemmas 2, 3 and definition, we have that ${\omega }_g\left(N\left[v_c\right]\right)\ge 4$ and ${\omega }_g\left(N\left[v_d\right]\right)\ge 4$. Since $N\left(v_c\right)\bigcap N\left(v_d\right)=\varnothing$, then we can reduce ${\omega }_g\left(T_v^{\prime }\right)={\omega }_g\left(N\left[v_c\right]\right)+{\omega }_g\left(N\left[v_d\right]\right)\ge 8$. □

Claim 2.

If there exists a $TR3DF$ h of $G^{\prime }$ with $h\left(v_a\right)+h\left(v_b\right)\ge 3$ for $v_a,v_b\in V\left(T_v\right)$, then there exists a $TR3DF$ g of $G^{\prime }$ such that ${\omega }_g\left(G^{\prime }\right)\le {\omega }_h\left(G^{\prime }\right)$ and $g\left(v_a\right)+g\left(v_b\right)\le 2$.

Proof of Claim 2. 

By the definition of $TR3DF$, we have ${\omega }_h\left(N\left[v_e\right]\right)\ge 3$ and ${\omega }_h\left(N\left[v_p\right]\right)\ge 3$, then we have ${\omega }_h\left(T_v^{\prime }\right)\ge 9$.

If $h\left(v\right)=0$, then we define $g:V\left(G^{\prime }\right)\to \{0,1,2,3\}$ such that $g\left(v_e\right)=g\left(v_f\right)=g\left(v_p\right)=g\left(v_q\right)=0$, $g\left(v\right)=g\left(v_a\right)=g\left(v_b\right)=1$, $g\left(v_c\right)=g\left(v_d\right)=3$, $g\left(x\right)=h\left(x\right)$ otherwise, seeing Figure 2. Therefore g is a $TR3DF$ of $G^{\prime }$ such that $g\left(v_a\right)+g\left(v_b\right)\le 2$ and ${\omega }_g\left(G^{\prime }\right)={\omega }_h\left(G^{\prime }\right)$.

If $h\left(v\right)\ge 1$, then we define $g:V\left(G^{\prime }\right)\to \{0,1,2,3\}$ such that $g\left(v_e\right)=g\left(v_f\right)=g\left(v_p\right)=g\left(v_q\right)=0$, $g\left(v_a\right)=g\left(v_b\right)=1$, $g\left(v_c\right)=g\left(v_d\right)=3$, $g\left(x\right)=h\left(x\right)$ otherwise. Therefore g is a $TR3DF$ of $G^{\prime }$ such that $g\left(v_a\right)+g\left(v_b\right)\le 2$ and ${\omega }_g\left(G^{\prime }\right)\le {\omega }_h\left(G^{\prime }\right)$. □

Lemma 5.

If g is a $TR3DF$ of G with ${\omega }_g\left(G^{\prime }\right)\le \ell +8n$, then there exists a $DF$ f of G with ${\omega }_f\left(G\right)\le \ell$.

Proof of Lemma 5. 

By Claim 2, w.l.o.g, let g be a $TR3DF$ of $G^{\prime }$ with $g\left(v_a\right)+g\left(v_b\right)\le 2$ for $v_a,v_b\in V\left(T_v\right)$, $v\in V\left(G\right)$. Define $f:V\left(G\right)\to \{0,1\}$ such that $f\left(v\right)=g\left(v\right)$ if $g\left(v\right)\le 1$, and $f\left(v\right)=1$ if $g\left(v\right)\ge 2$. For each vertex $v\in V\left(G\right)$, since $g\left(v_a\right)+g\left(v_b\right)\le 2$, we have $g\left(v\right)\ge 1$ or there exists a vertex $u\in N\left(v\right)\cap V\left(G\right)$ such that $g\left(u\right)\ge 1$. Therefore f is $DSF$ of G and ${\omega }_f\left(G\right)\le {\omega }_g\left(G\right)-8n\le \ell$ by Claim 1. □

Theorem 1.

By Lemmas 1, 4, 5, the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs.

3. A Linear-Time Algorithm for Total Roman {3}-Domination in Trees

In this section, we present a linear-time algorithm to compute the minimum weight of total Roman $\{3\}$-dominating function for trees. First, we define the following concepts:

Definition 1.

Let u be a vertex of G, and let $F_{u,G}^{(i,j)}$ on G be a function $f:V\left(G\right)\to \{0,1,2,3\}$ having the property that (i) $f\left(u\right)=i$, ${\sum }_{w\in N\left(u\right)}f\left(w\right)\ge j$; (ii) $\forall v\in V\left(G\right)\backslash \{u\}$, ${\sum }_{p\in N\left[v\right]}f\left(p\right)\ge 3$ if $f\left(v\right)\le 2$ and ${\sum }_{p\in N\left(v\right)}f\left(p\right)\ge 1$ if $f\left(v\right)=3$.

Definition 2.

The minimum weight overall $F_{u,G}^{(i,j)}$ functions on G denoted by ${\gamma }_{tR3}^{(i,j)}(u,G)$ is the $F_{u,G}^{(i,j)}$ number of G, and a ${\gamma }_{tR3}^{(i,j)}(u,G)$-function is an $F_{u,G}^{(i,j)}$ function on G with weight ${\gamma }_{tR3}^{(i,j)}(u,G)$.

Definition 3.

Let $coil\left(x\right)$ be a function defined as follows: $coil\left(x\right)$ = $\left\{\begin{array}{c}x,x\ge 0;\hfill \\ 0,x<0.\hfill \end{array}\right.$

Lemma 6.

For any graph G with specific vertex u, we have

$${\gamma }_{t\{R3\}}\left(G\right)=min\{{\gamma }_{tR3}^{(0,3)}(u,G),{\gamma }_{tR3}^{(1,2)}(u,G),{\gamma }_{tR3}^{(2,1)}(u,G),{\gamma }_{tR3}^{(3,1)}(u,G)\}.$$

Lemma 7.

Suppose $T_1$ and $T_2$ are trees with specific vertices v and u, respectively. Let $T_3$ be the tree with the specific vertex u, which is obtained by joining a new edge $uv$ from the union of $T_1$ and $T_2$, as depicted in Figure 3.

Then the following statements hold for ${\gamma }_{tR3}^{(i,j)}(u,T_k)$.

$\left(a\right)$ For $i=0$, $j\in \{0,1,2,3\}$, we have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\gamma }_{tR3}^{(0,j)}(u,T_3)=min\{& {\gamma }_{tR3}^{(3,1)}(v,T_1)+{\gamma }_{tR3}^{(0,0)}(u,T_2),\hfill \\ & min\{{\gamma }_{tR3}^{(s,3-s)}(v,T_1)+{\gamma }_{tR3}^{(0,coli(j-s\left)\right)}(u,T_2)|s=0,1,2\}\}\hfill \end{array}\end{array}$$

$\left(b\right)$ For $i\in \{1,2,3\}$, $j\in \{0,1,2,3\}$, we have:

$$\begin{array}{c}\hfill {\gamma }_{tR3}^{(i,j)}(u,T_3)=min\{{\gamma }_{tR3}^{(s,coil(3-i-s\left)\right)}(v,T_1)+{\gamma }_{tR3}^{(i,coil(j-s\left)\right)}(u,T_2)|s=0,1,2,3\}\end{array}$$

Proof of Lemma 7. 

Let $V\left(T_1^{\prime }\right)=V\left(T_1\right)\cup \{u\}$, $E\left(T_1^{\prime }\right)=E\left(T_1\right)\cup \{vu\}$, f be a ${\gamma }_{tR3}^{(i,j)}(u,G)$-function of $T_3$, $f^{\prime }$ be the restriction of f on $T_1^{\prime }$ and $f^{″}$ be the restriction of f on $T_2$.

(a) If f is a ${\gamma }_{tR3}^{(0,j)}(u,T_3)$-function on $T_3$, for $j\in \{0,1,2,3\}$. By the definition of ${\gamma }_{tR3}^{(i,j)}(u,G)$-function, we have that if $f\left(v\right)=3$, then ${\sum }_{w\in N_{T_3\backslash \{u\}}}f\left(w\right)\ge 1$. It follows from the fact that f is a ${\gamma }_{tR3}^{(0,j)}(u,G)$-function of $T_3$ if and only if $f=f^{″}\cup f^{\prime }$, where at least one of followings holds: (i) $f^{″}$ is a ${\gamma }_{tR3}^{(0,0)}(u,G)$-function of $T_2$, $f^{\prime }$ is a ${\gamma }_{tR3}^{(3,1)}(v,T_1)$-function of $T_1$; (ii) $f^{″}$ is a ${\gamma }_{tR3}^{(0,coil(j-s\left)\right)}(u,G)$-function of $T_2$, $f^{\prime }$ is a ${\gamma }_{tR3}^{(s,3-s)}(v,T_1)$-function of $T_1$, for $s\in \{0,1,2\}$.

(b) It follows from the fact that f is a ${\gamma }_{tR3}^{(i,j)}(u, T_3)$-function of $T_3$, for $i\in \{1,2,3\}$, $j\in \{0,1,2,3\}$ if and only if f = $f^{″}\cup f^{\prime }$, where $f^{″}$ is a ${\gamma }_{tR3}^{(i,coli(j-s\left)\right)}(u,T_2)$-function of $T_2$ and $f^{\prime }$ is a ${\gamma }_{tR3}^{(t,coil(3-i-s\left)\right)}(v,T_1)$-function of $T_1$, for $s\in \{0,1,2,3\}$. □

Lemmas 6 and 7 give the following dynamic programming algorithm 1 for the total Roman $\{3\}$-domination problem in trees.

Algorithm 1 Counting ${\gamma }_{t\{R3\}}$ in trees.

4. Conclusions

The total Roman {3}-domination problem was introduced and studied in [24], and it was proven to be NP-complete for bipartite graphs. In this paper, we prove that the total Roman {3}-domination problem is NP-complete for planar graphs or chordal bipartite graphs, and showed a linear-time algorithm for total Roman {3}-domination problem on trees. For the algorithmic aspects of the total Roman {3}-domination problem, designing exact algorithms or approximation algorithms on general graphs, or polynomial algorithms for total Roman {3}-domination problem on some special classes graphs deserve further research.