Equitable restrained domination number of some graphs

Abstract

A dominating set $S \subseteq V$ is said to be a restrained dominating set of graph $G$ if every vertex not in $S$ is adjacent to a vertex in $S$ and also to a vertex in $V-S$. A set $S \subseteq V$ is called an equitable dominating set if for every vertex $v \in V-S$, there exist a vertex $u \in S$ such that $u v \in E$ and $|\operatorname{deg}(u)-\operatorname{deg}(v)| \leq 1$. A dominating set $S$ is called an equitable restrained dominating set if it is both restrained and equitable. The minimum cardinality of an equitable restrained dominating set is called equitable restrained domination number of $G$, denoted by $\gamma_r^e(G)$. We investigate $\gamma_r^f(G)$ parameter for some standard graphs and also establish some characterizations.