EQUITABLE RESOLVING DOMINATING SETS IN GRAPHS

Abstract

A dominating set $D \subseteq V(G)$ is called an equitable dominating set if for every vertex $v \in V(G)-D$, there exists a vertex $u \in D$ such that $u v \in E(G)$ and $|\operatorname{deg}(u)-\operatorname{deg}(v)| \leq 1$. The distance $d(u, v)$ between two vertices in $G$ is the length of the shortest path between $u$ and $v$ in $G$. Let $W=\left\{w_1, w_2, \ldots, w_k\right\}$ be an ordered subset of $V(G)$ and let $v \in V(G)$. Then the $k$-vector $\left(d\left(v, w_1\right), d\left(v, w_2\right), \ldots, d\left(v, w_k\right)\right)$ is called the resolving vector of $v$ with respect to $W$ and is denoted by $r(v \mid W)$. The set $W$ is called a resolving set of $G$ if $r(u \mid W) \neq r(v \mid W)$ for any two distinct vertices $u$ and $v$. A set $R_D \subseteq V(G)$ is called a resolving dominating set if it is resolving and dominating both. A dominating set $D$ is called an equitable resolving dominating set if it is resolving as well as equitable. The minimum cardinality of an equitable resolving dominating set is called an equitable resolving domination number of $G$, denoted by $\gamma_{r s}^e(G)$. In the present work, some characterizations are established and equitable resolving domination numbers for various graphs are investigated.

Received: September 26, 2022;
Revised: February 2, 2023;
Accepted: February 14, 2023;