On the Double Roman Domination Number of Generalized Sierpinski Graphs

TL;DR

This paper investigates the double Roman domination number in generalized Sierpinski graphs, providing bounds and exact values for specific cases, advancing understanding of domination parameters in fractal-like graph structures.

Contribution

It offers new bounds for the double Roman domination number of generalized Sierpinski graphs and determines the exact value for a specific case, filling gaps in graph domination theory.

Findings

Established bounds for $ ext{double Roman domination}$ in $S(G,t)$

Calculated exact $ ext{double Roman domination number}$ for $S(K_{n}, 2)$

Enhanced understanding of domination parameters in fractal graphs

Abstract

In this paper, we study the double Roman domination number of generalized Sierpi\'{n}ski graphs $S(G,t)$. More precisely, we obtain a bound for the double Roman domination number of $S(G, t)$. We also find the exact value of $\gamma_{dR}(S(K_{n}, 2))$.

Equations

$${\gamma }_R(S(K_n,t))⩽\{\begin{array}{ll}\frac{2n^t+n-1}{n+1},& \text{t even},\\ \frac{2(n^t+1)}{n+1},& \text{t odd}.\end{array}$$

$$n^{t-2}\alpha (G){\gamma }_{dR}(G)⩽{\gamma }_{dR}(S(G,t))⩽n^{t-2}(n{\gamma }_{dR}(G)-|V_3|-|D_3|),$$

Full text

Keywords: Double Roman Dominating Function, Double Roman Domination Number, Sierpiński Graphs.

AMS Subject Classification: 05C69; 05C76

1 Introduction

Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. If there is no ambiguity in the choice of $G$, then we write $V(G)$ and $E(G)$ as $V$ and $E$, respectively. Let $f:V\rightarrow\{0,1,2,3\}$ be a function defined on $V(G).$ Let $V_{i}^{f}=\{v\in V(G):f(v)=i\}.$ (If there is no ambiguity, $V_{i}^{f}$ is written as $V_{i}$.) Then $f$ is a double Roman dominating function (DRDF) on $G$ if it satisfies the following conditions.

(i) If $v\in V_{0}$, then vertex $v$ must have at least two neighbors in $V_{2}$ or at least one neighbor in $V_{3}$.

(ii) If $v\in V_{1}$, then vertex $v$ must have at least one neighbor in $V_{2}\cup V_{3}$.

The weight of a DRDF $f$ is the sum $f(V)=\sum_{v\in V}f(v)$. The double Roman domination number, $\gamma_{dR}(G)$, is the minimum among the weights of DRDFs on $G$, and a DRDF on $G$ with weight $\gamma_{dR}(G)$ is called a $\gamma_{dR}$-function of $G$ [6].

Let $(V_{0},V_{1},V_{2},V_{3})$ be the ordered partition of $V$ induced by $f$. Note that there exists a $1-1$ correspondence between the functions $f$ and the ordered partitions $(V_{0},V_{1},V_{2},V_{3})$ of $V$. Thus we will write $f=(V_{0},V_{1},V_{2},V_{3})$.

R. A. Beeler, T. W. Haynes and S. T. Hedetniemi pioneered the study of double Roman domination in [6]. The relationship between double Roman domination and Roman domination and the bounds on the double Roman domination number of a graph $G$ in terms of its domination number were discussed by them. They also determined a sharp upper bound on $\gamma_{dR}(G)$ in terms of the order of $G$ and characterized the graphs attaining this bound. In [1], it was verified that the decision problem associated with $\gamma_{dR}(G)$ is NP-complete for bipartite and chordal graphs. Above all this, a characterization of graphs $G$ with small $\gamma_{dR}(G)$ was provided. In [8], G. Hao et al. introduced the study of the double Roman domination of digraphs and L. Volkmann proposed a sharp lower bound on $\gamma_{dR}(G)$ in [12]. In [3], it was proved that $\gamma_{dR}(G)+2\leqslant\gamma_{dR}(M(G))\leqslant\gamma_{dR}(G)+3$, where $M(G)$ is the Mycielskian graph of $G$ and a construction was also given which confirms that there is no relation between the double Roman domination number of a graph and its induced subgraphs. The impact of some graph operations such as corona, cartesian product and addition of twins, on double Roman domination number was studied in [4]. In [2], J. Amjadi et al. improved the upper bound on $\gamma_{dR}(G)$ given in [6] by showing that for any connected graph $G$ of order $n$ with minimum degree at least two, $\gamma_{dR}(G)\leqslant\frac{8n}{7}$.

1.1 Basic Definitions and Preliminaries

The open neighborhood of a vertex $v\in V$ is the set $N(v)=\{u:uv\in E\}$, and its closed neighborhood is $N[v]=N(v)\cup\{v\}$. The vertices in $N(v)$ are called the neighbors of $v$. When $G$ must be explicit, these open and closed neighborhoods are denoted by $N_{G}(v)$ and $N_{G}[v]$, respectively. $|N(v)|$ is called the degree of the vertex $v$ in $G$ and is denoted by $d_{G}(v)$, or simply $d(v)$. A vertex of degree [math] is known as an isolated vertex of $G$.

If $U$ is a non-empty subset of the vertex set $V$ of the graph $G$ then the subgraph $<U>$ of $G$ induced by $U$ is defined as the graph having vertex set $U$ and edge set consisting of those edges of $G$ that have both ends in $U$. A subset $S$ of the vertex set $V$ of a graph $G$ is called independent if no two vertices of $S$ are adjacent in $G$. $S\subseteq V$ is a maximum independent set of $G$ if $G$ has no independent set $S^{\prime}$ with $|S^{\prime}|>|S|$. The number of vertices in a maximum independent set of $G$ is called the independence number, denoted by $\alpha(G)$. A complete graph on $n$ vertices, denoted by $K_{n}$, is the graph in which any two vertices are adjacent.

A Roman dominating function (RDF) on a graph $G=(V,E)$ is defined as a function $f:V\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $v$ for which $f(v)=0$ is adjacent to at least one vertex $u$ for which $f(u)=2$. The weight of a RDF is the value $f(V)=\sum_{v\in V}f(v)$. The Roman domination number of a graph $G$, denoted by $\gamma_{R}(G)$, is the minimum among the weights of RDFs on $G$.

Let $G=(V,E)$ be a non-empty graph of order $n\geqslant 2$, and $t$ a positive integer. We denote by $V^{t}$ the set of words of length $t$ on alphabet $V$. The letters of a word $u$ of length $t$ are denoted by $u_{1}u_{2}\ldots u_{t}$. Klavžar and Milutinović introduced in [9] the graph $S(K_{n},t),t\geqslant 1$, ($S(t,n)$ in their notation) whose vertex set is $V^{t},$ where $\{u,v\}$ is an edge if and only if there exists $i\in\{1,2,\ldots,t\}$ such that:

(i) $u_{j}=v_{j},$ if $j<i$; (ii) $u_{i}\neq v_{i}$; (iii) $u_{j}=v_{i}$ and $v_{j}=u_{i}$ if $j>i$.

Later, those graphs have been called Sierpiński graphs in [10]. This construction was generalized in [7] for any graph $G=(V,E)$, by defining the $t^{th}$ generalized Sierpiński graph of $G$, denoted by $S(G,t)$, as the graph with vertex set $V^{t}$ and edge set $\{\{wu_{i}u_{j}^{r-1},wu_{j}u_{i}^{r-1}\}:\{u_{i},u_{j}\}\in E,i\neq j;r\in\{1,2,\ldots,t\};w\in V^{t-r}\}$. (See Figure 1 and 2.)

Vertices of the form $xx\ldots x$ are called extreme vertices of $S(G,t)$. Note that for any graph $G$ of order $n$ and any integer $t\geqslant 2$, $S(G,t)$ has $n$ extreme vertices and, if $x$ has degree $d(x)$ in $G$, then the extreme vertex $xx\ldots x$ of $S(G,t)$ also has degree $d(x)$.

For any graph theoretic terminology and notations not mentioned here, the readers may refer to [5]. The following results are useful in this paper.

Proposition 1.1**.**

[6]** In a double Roman dominating function of weight $\gamma_{dR}(G)$, no vertex needs to be assigned the value $1$.

i.e., For any graph $G$, there exists a $\gamma_{dR}$-function with $V_{1}=\emptyset$.

Theorem 1.2**.**

[11]** For any integers $n\geqslant 2$ and $t\geqslant 1$,

[TABLE]

2 Bounds on the Double Roman Domination Number

First we prove a lower bound for $\gamma_{R}(S(G,t))$.

Theorem 2.1**.**

For any graph $G$ of order $n$, $\gamma_{R}(S(G,t))\geqslant n^{t-2}\alpha(G)\gamma_{R}(G)$, where $\alpha(G)$ is the independence number of $G$.

Proof.

Let $V^{\prime}\subseteq V$ be an independent set of cardinality $\alpha(G)$. For any $w\in V^{t-2},i\in V,$ let $V_{wi}=\{wij:j\in V\}.$ Note that $\{V_{wi}:i\in V\}$ is a partition of the vertex set of $S(G,t)$ and $<V_{wi}>\cong G,$ for every $i$ and hence there are $n^{t-1}$ disjoint copies of $G$ in $S(G,t).$ If $u$ and $v$ are adjacent in $S(G,t),$ then $u$ and $v$ are of the form $u=wxy^{r-1},v=wyx^{r-1},$ where $w\in V^{t-r},r\in\{1,2,\ldots,t\}$ and $x$ and $y$ are adjacent in $G.$ Hence, for every $i,j\in V^{\prime},$ none of the vertices in $V_{wi}$ is adjacent to any of the vertices in $V_{wj}.$ Also, $N(V_{wi})\cap N(V_{wj})=\emptyset,$ for $i\neq j.$ Therefore, $f(V^{t})\geqslant n^{t-2}\alpha(G)\gamma_{R}(G)$ for any RDF $f$ of $S(G,t)$ and hence $\gamma_{R}(S(G,t))\geqslant n^{t-2}\alpha(G)\gamma_{R}(G).$ ∎

Remark 2.1**.**

It is clear that the inequality in Theorem 2.1 holds for other domination parameters like domination number, independence domination number, total domination number and many more.

Theorem 2.2**.**

Let $G$ be a graph of order $n$. For any $\gamma_{dR}$-function $f=(V_{0},V_{2},V_{3})$ on $G$, and any integer $t\geqslant 2$,

[TABLE]

where $\alpha(G)$ is the independence number of $G$ and $D_{3}$ is the set of non-isolated vertices in $<V_{3}>$.

Proof.

For the left inequality, the proof is as same as that of the Theorem 2.1. To prove the right inequality, let $f=(V_{0},V_{2},V_{3})$ be a $\gamma_{dR}$-function on $G$.

Step 1: For a given integer $t\geqslant 2$, let $S_{i}=\{wx:w\in V^{t-1},x\in V_{i}\},$ for $i\in\{0,2,3\}.$ Let $g:V^{t}\rightarrow\{0,2,3\}$ such that $g=(S_{0},S_{2},S_{3}).$ If $v\in V^{t}$ and $g(v)=0$, then $v=wy$ where $w$ is a word in $V^{t-1}$ and $y\in V_{0}$. Since $f$ is a $\gamma_{dR}$-function on $G$, there is either $z\in V_{3}\cap N_{G}(y)$ or $x_{1},x_{2}\in V_{2}\cap N_{G}(y).$ Hence, there exists either $wz\in V_{3}\cap N_{S(G,t)}(wy)$ or $wx_{1},wx_{2}\in V_{2}\cap N_{S(G,t)}(wy).$ Therefore, $g$ is a double Roman dominating function on $S(G,t)$ and $\gamma_{dR}(S(G,t))\leqslant g(V^{t})=n^{t-1}(2|V_{2}|+3|V_{3}|)=n^{t-1}\gamma_{dR}(G).$

Step 2: Let $S_{3}^{\prime}=\{wuu:w\in V^{t-2},u\in V_{3}\}.$ We define $g_{1}:V^{t}\rightarrow\{0,2,3\}$ such that $g_{1}=(S_{0},S_{2}\cup S_{3}^{\prime},S_{3}-S_{3}^{\prime}).$ Let $y\in S_{0}.$ Then $y$ has the form $wuv_{0}$ where $w\in V^{t-2},u\in V$ and $v_{0}\in V_{0}.$ Since $f$ is a $\gamma_{dR}$-function on $G,$ there is either $v_{3}\in V_{3}$ or $v_{2},v_{2}^{\prime}\in V_{2}$ in $N_{G}(v_{0}).$ Therefore, there exists either $wuv_{3}\in N_{S(G,t)}(wuv_{0})$ or $wuv_{2},wuv_{2}^{\prime}\in N_{S(G,t)}(wuv_{0}).$ If $wuv_{3}\in S_{3}-S_{3}^{\prime},$ or $wuv_{2},wuv_{2}^{\prime}\in N_{S(G,t)}(wuv_{0}),$ then we are done. Now, if $wuv_{3}\in S_{3}^{\prime},$ then $v_{3}=u$ and, since $v_{0}$ is adjacent to $v_{3},$ we can conclude that $y=wv_{3}v_{0}$ is adjacent to $wv_{0}v_{3}\in S_{3}-S_{3}^{\prime}.$ Hence $g_{1}$ is a double Roman dominating function on $S(G,t)$. Therefore, $\gamma_{dR}(S(G,t))\leqslant g_{1}(V^{t})\leqslant n^{t-2}(n\gamma_{dR}(G)-|V_{3}|).$

Step 3: Let $S_{3}^{\prime\prime}=\{wvv:w\in V^{t-2},v\in D_{3},where\ D_{3}\ is\ the\ set\ of\ non-isolated\ vertices\ in\ <V_{3}>\}.$ We define $g_{2}:V^{t}\rightarrow\{0,2,3\}$ such that $g_{2}=(S_{0}\cup S_{3}^{\prime\prime},S_{2}\cup S_{3}^{\prime},S_{3}-S_{3}^{\prime}).$ Let $x\in V^{t}$ such that $g_{2}(x)=0.$ In this case, $g_{1}(x)=0$ or $x\in S_{3}^{\prime\prime}.$

Suppose that $g_{1}(x)=0.$ Then $x\in S_{0}$ and hence is of the form $x=wuv_{0},$ where $w\in V^{t-2},u\in V$ and $v_{0}\in V_{0}.$ If $N_{S(G,t)}(x)\cap S_{3}^{\prime\prime}=\emptyset,$ then there exists $y\in N_{S(G,t)}(x)\cap(S_{3}-S_{3}^{\prime})$ or $y_{1},y_{2}\in N_{S(G,t)}(x)\cap(S_{2}\cup S_{3}^{\prime}).$ On the other side, if $z\in N_{S(G,t)}(x)\cap S_{3}^{\prime\prime},$ then $z=wv_{3}v_{3},$ where $v_{3}\in D_{3},$ $u=v_{3}$ and $v_{3}$ is adjacent to $v_{0},$ which implies that $x=wv_{3}v_{0}$ is adjacent to $wv_{0}v_{3}$ and we know that $g_{2}(wv_{0}v_{3})=g_{1}(wv_{0}v_{3})=g(wv_{0}v_{3})=3.$

Now, if $x\in S_{3}^{\prime\prime},$ then $x=wvv,$ where $w\in V^{t-2}$ and $v\in D_{3}.$ So, by definition of $D_{3},$ $x$ must be adjacent to $wvu$ for some $u\in D_{3}-\{v\}.$ Also, $g_{2}(wvu)=g_{1}(wvu)=g(wvu)=f(u)=3.$ Therefore, $g_{2}$ is a double Roman dominating function on $S(G,t),$ and so $\gamma_{dR}(S(G,t))\leqslant n^{t-2}(n\gamma_{dR}(G)-|V_{3}|-|D_{3}|).$ ∎

3 The Particular Case of Complete Graphs

We begin this section by proving the Roman domination number of $S(K_{n},2)$.

Theorem 3.1**.**

$\gamma_{R}(S(K_{n},2))=2n-1$.

Proof.

By Theorem 1.2, we can easily deduce that $\gamma_{R}(S(K_{n},2))\leqslant 2n-1$. For the reverse inequality, let $V(K_{n})=\{u_{1},u_{2},\ldots,u_{n}\}$. Then $S(K_{n},2)$ is a graph with vertex set $\{u_{i}u_{j}:i,j\in\{1,2,\ldots,n\}\}$ and edge set $\{\{u_{i}u_{j},u_{j}u_{i}\}:u_{i},u_{j}\in V,i\neq j\}\cup\{\{u_{i}u_{j},u_{i}u_{k}\}:u_{i},u_{j},u_{k}\in V,j\neq k\}$. Note that $G_{i}=\{u_{i}u_{j}:j=1,2,\ldots,n\}$ induces a copy of $K_{n}$ for each $i=1,2,\ldots,n$ and vertex $u_{i}u_{i}$ is an extreme vertex of $S(K_{n},2)$ for each $i.$ Let $f=(V_{0},V_{1},V_{2})$ be any $\gamma_{R}$-function of $S(K_{n},2).$ Since $d_{S(K_{n},2)}(u_{i}u_{i})=d_{K_{n}}(u_{i})=n-1$, to Roman dominate the extreme vertex $u_{i}u_{i}$, $G_{i}$ must contain either a vertex of value $2$ or $u_{i}u_{i}$ is of value $1$. If every $G_{i}$ contains a vertex of value $2$, then $f(V(S(K_{n},2)))\geqslant 2n$, which is a contradiction. Therefore, there exists at least one $G_{i},$ say $G_{i_{0}},$ which contains exactly one vertex in $V_{1}$ (and all other vertices are in $V_{0}$). Then, by the property of extreme vertex, the vertex in $V_{1}$ is $u_{i_{0}}u_{i_{0}}.$ So, to Roman dominate $u_{i_{0}}u_{j},$ for each $j\neq i_{0},$ $u_{j}u_{i_{0}}\in V_{2}.$ Therefore, $\gamma_{R}(S(K_{n},2))\geqslant 2(n-1)+2=2n-1.$ Hence, the result. ∎

Theorem 3.2**.**

$\gamma_{dR}(S(K_{n},2))=3n-1$.

Proof.

By Theorem 2.2, we can easily deduce that $\gamma_{dR}(S(K_{n},2))\leqslant 3n-1$, since $|V_{3}|=1$ and $|D_{3}|=0$. For the reverse inequality, let $V(K_{n})=\{u_{1},u_{2},\ldots,u_{n}\}$. Then $S(K_{n},2)$ is a graph with vertex set $\{u_{i}u_{j}:i,j\in\{1,2,\ldots,n\}\}$ and edge set $\{\{u_{i}u_{j},u_{j}u_{i}\}:u_{i},u_{j}\in V,i\neq j\}\cup\{\{u_{i}u_{j},u_{i}u_{k}\}:u_{i},u_{j},u_{k}\in V,j\neq k\}$. Note that $G_{i}=\{u_{i}u_{j}:j=1,2,\ldots,n\}$ induces a copy of $K_{n}$ for each $i=1,2,\ldots,n$ and vertex $u_{i}u_{i}$ is an extreme vertex of $S(K_{n},2)$ for each $i.$ Let $f$ be any $\gamma_{dR}$-function of $S(K_{n},2).$ To double Roman dominate the extreme vertex $u_{i}u_{i},$ $G_{i}$ must contain either one vertex having value $3,$ or two vertices having value $2$ or $u_{i}u_{i}$ is of value $2.$ If every $G_{i}$ contains a vertex of value $3$ or two vertices of value 2, then $f(V(S(K_{n},2))\geqslant 3n$ which is a contradiction to Theorem 2.2. Therefore, there exists at least one $G_{i},$ say $G_{i_{0}},$ which contains exactly one vertex having value $2.$ Then, by the property of extreme vertex the value $2$ is assigned to $u_{i_{0}}u_{i_{0}}.$ To double Roman dominate $u_{i_{0}}u_{j},$ for each $j\neq i_{0},$ $u_{j}u_{i_{0}}$ must be in $V_{2}\cup V_{3}.$ If $u_{j}u_{i_{0}}\in V_{2},$ then $G_{j}$ contains at least one more vertex of value at least $2.$ Therefore, it is optimal to assign the value $3$ to $u_{j}u_{i_{0}},$ for every $j\neq i_{0}.$ But then $f(V(S(K_{n},2))\geqslant 2+3(n-1).$ Hence, $\gamma_{dR}(S(K_{n},2))=3n-1.$ ∎

Acknowledgements: The first author thanks University Grants Commission for granting fellowship under Faculty Development Programme (F.No.FIP/$12^{th}$ plan/KLMG 045 TF 07 of UGC-SWRO).