Perfect double Roman domination of trees

Abstract

For a graph $G$ with vertex set $V\left(G\right)$ and function $f:V\left(G\right)\to \{0,1,2,3\}$, let $V_i$ be the set of vertices assigned $i$ by $f$. A perfect double Roman dominating function of a graph $G$ is a function $f:V\left(G\right)\to \{0,1,2,3\}$ satisfying the conditions that (i) if $u\in V_0$, then $u$ is either adjacent to exactly two vertices in $V_2$ and no vertex in $V_3$ or adjacent to exactly one vertex in $V_3$ and no vertex in $V_2$; and (ii) if $u\in V_1$, then $u$ is adjacent to exactly one vertex in $V_2$ and no vertex in $V_3$. The perfect double Roman domination number of $G$, denoted ${\gamma }_{dR}^p\left(G\right)$, is the minimum weight of a perfect double Roman dominating function of $G$. We prove that if $T$ is a tree of order $n\ge 3$, then ${\gamma }_{dR}^p\left(T\right)\le 9n∕7$. In addition, we give a family of trees $T$ of order $n$ for which ${\gamma }_{dR}^p\left(T\right)$ approaches this upper bound as $n$ goes to infinity.

Introduction

Paramount to the Roman army were its legions, made up of highly trained solders. These legions were stationed at various locations in the empire to defend Rome from attacks by neighboring countries. A location is protected by a legion stationed there, while a location having no legion can be protected by a legion sent from a neighboring location. In an attempt to protect all cities while preventing the problem of leaving a location unprotected when its legion is dispatched to defend another, Emperor Constantine decreed that a city having no legions must be within the vicinity of at least one location having two legions, one of which could be dispatched to the attacked city. For economical reasons, he further decreed that no more than two legions be stationed at any location. Motivated by a series of articles outlining Roman defense strategies in mathematical magazines (see [22], [23], [24], [25]), Cockayne, et al. [7] introduced Roman domination as a graph theoretical concept as follows.

Let $f$ be a function assigning a non-negative integer to each vertex of $V\left(G\right)$. The weight of a vertex $v$ is its value $f\left(v\right)$ assigned to it under $f$. The weight $\mathrm{w}\left(f\right)$ of $f$ is the sum, ${\sum }_{u\in V\left(G\right)}f\left(u\right)$, of the weights of the vertices. Let $V_i$ denote the set of vertices assigned $i$ by function $f$.

A Roman dominating function on $G$ is a function $f:V\left(G\right)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $f\left(u\right)=0$ is adjacent to at least one vertex $v$ for which $f\left(v\right)=2$. The Roman domination number, denoted ${\gamma }_R\left(G\right)$, is the minimum weight of an Roman dominating function of $G$. Since its introduction in graph theory in 2004, Roman domination has been the topic of over 100 research papers. We refer the reader to [5], [6], [8], [11], [15], [18], [19], [20], [21], [27] for a small sample of recent papers on Roman domination and its variants.

Double Roman domination, a variant of Roman domination, was proposed by Beeler et al. [4] as a stronger version of Roman domination. As we have seen, the strategy of Roman domination provides for any attacked location to be defended by one legion. Double Roman domination doubles the protection by ensuring that any attack can be defended by at least two legions. A function $f:V\left(G\right)\to \{0,1,2,3\}$ is a double Roman dominating function on a graph $G$ if the following conditions are met.

• If $v\in V_0$, then vertex $v$ has at least two neighbors in $V_2$ or at least one neighbor in $V_3$.

• If $v\in V_1$, then vertex $v$ has at least one neighbor in $V_2\cup V_3$.

The double Roman domination number, denoted ${\gamma }_{dR}\left(G\right)$, is the minimum weight of a double Roman dominating function of $G$. It is noted in [4] that double Roman domination, with the ability to deploy three legions at a given location, allows to double the level of defense, sometimes at less than the anticipated additional cost. For example, consider the star $K_{1,n-1}$, for which ${\gamma }_{dR}\left(K_{1,n-1}\right)=3<4=2{\gamma }_R\left(K_{1,n-1}\right)$. For more details on double Roman domination, see [1], [2], [3], [14], [16], [26], [28], [29].

A Roman dominating function is called perfect in [12] if for every vertex $v$ with $f\left(v\right)=0$, there is exactly one vertex $u\in N\left(v\right)$ with $f\left(u\right)=2$. The authors in [12] showed that if $T$ is a tree on $n\ge 3$ vertices, then ${\gamma }_R^p\left(T\right)\le \frac{4}{5}n$, and characterized the trees achieving equality in this bound. For another example of a perfect function involving a variant of Roman domination, see [10]. See also Klostermeyer’s taxonomy of perfect domination [17].

We extend the concept of perfect to double Roman domination as follows. A perfect double Roman dominating function of a graph $G$, abbreviated PDRD-function, is a function $f$ : $V\left(G\right)\to \{0,1,2,3\}$ satisfying the following conditions:

• If $u\in V_0$, then either $u$ is adjacent to exactly two vertices in $V_2$ and no vertex in $V_3$, or $u$ is adjacent to exactly one vertex in $V_3$ and no vertex in $V_2$.

• If $u\in V_1$, then $u$ is adjacent to exactly one vertex in $V_2$ and no vertex in $V_3$.

The perfect double Roman domination number of $G$, denoted ${\gamma }_{dR}^p\left(G\right)$, is the minimum weight among all PDRD-functions of $G$. That is,

$${\gamma }_{dR}^p\left(G\right)=min\{\mathrm{w}\left(f\right)\mid f\text{ is a PDRD-function in }G\}.$$

A PDRD-function of $G$ with weight ${\gamma }_{dR}^p\left(G\right)$ is called a ${\gamma }_{dR}^p$-$function$ of $G$.

Since every PDRD-function is a double Roman dominating function, ${\gamma }_{dR}\left(G\right)\le {\gamma }_{dR}^p\left(G\right)$ for any graph $G$. We note that there exists a ${\gamma }_{dR}$-function of a path $P_n$ that is also a PDRD-function, so we have ${\gamma }_{dR}^p\left(P_n\right)={\gamma }_{dR}\left(P_n\right)$. The following value of ${\gamma }_{dR}\left(P_n\right)$, and hence for ${\gamma }_{dR}^p\left(P_n\right)$, is given in [1].

Proposition 1

For the paths $P_n$,

$${\gamma }_{dR}^p\left(P_n\right)=\left\{\begin{array}{cc}n\hfill & \text{if }n\equiv 0\text{ mod 3},\hfill \\ n+1\hfill & \text{otherwise.}\hfill \end{array}\right.$$

For notation and graph theory terminology, we in general follow the books [9], [13]. Specifically, let $G=(V,E)$ be a graph with vertex set $V=V\left(G\right)$ of order $n=\left|V\right|$ and edge set $E=E\left(G\right)$, and let $v$ be a vertex in $V$. The open neighborhood of $v$ is the set $N_G\left(v\right)=\{u\in V|uv\in E\}$ and the closed neighborhood of $v$ is $N_G\left[v\right]=\left\{v\right\}\cup N_G\left(v\right)$. The degree of $v$ in $G$ is ${deg}_G\left(v\right)=\left|N_G\left(v\right)\right|$. The minimum degree among the vertices of $G$ is denoted by $\delta \left(G\right)$. If the graph $G$ is clear from the context, we omit it in the above expressions. For example, we write $n$, $d\left(u\right)$, $N\left(v\right)$ and $N\left[v\right]$ rather than $n\left(G\right)$, $d_G\left(u\right)$, $N_G\left(v\right)$ and $N_G\left[v\right]$, respectively. A leaf is a vertex of degree $1$, while its neighbor is a support vertex. We use the standard notation $\left[k\right]=\{1,\dots ,k\}$.

The distance $d_G(u,v)$ between two vertices $u$ and $v$ in a connected graph $G$ is the length of a shortest $(u,v)$-path in $G$. The maximum distance among all pairs of vertices of $G$ is the diameter of $G$, denoted by $\mathrm{diam}\left(G\right)$.

A rooted tree $T$ distinguishes one vertex $r$ called the root. For each vertex $v\ne r$ of $T$, the parent of $v$ is the neighbor of $v$ on the unique $(r,v)$-path, while a child of $v$ is any other neighbor of $v$ in $T$. A descendant of $v$ is a vertex $u\ne v$ such that the unique $(r,u)$-path contains $v$. In particular, every child of $v$ is a descendant of $v$. A grandchild of $v$ is the descendant of $v$ at distance $2$ from $v$. We let $D\left(v\right)$ denote the set of descendants of $v$, and we define $D\left[v\right]=D\left(v\right)\cup \left\{v\right\}$. The maximal subtree at $v$ is the subtree of $T$ induced by $D\left[v\right]$, and is denoted by $T_v$. The tree obtained from a rooted tree $T$ by deleting a subtree $T_v$ is denoted by $T-T_v$. For $r,s\ge 1$, a double star $S(r,s)$ is the tree with exactly two vertices that are not leaves, one of which has $r$ leaf neighbors and the other $s$ leaf neighbors.