SOME RESULTS ON 1-MOVABLE RESTRAINED PERFECT DOMINATING SETS IN THE JOIN AND CORONA OF GRAPHS

Abstract

A nonempty subset $S$ of $V(G)$ is a 1-movable restrained perfect dominating set of $G$ if $S=V(G)$ or $\mathrm{S} \subset \mathrm{V}(\mathrm{G})$ is a restrained perfect dominating set of $G$ and for every $v \in S$, there exists $u \in(V(G) \backslash S) \cap N_G(v)$ such that $(S \backslash\{v\}) \bigcup\{u\}$ is a restrained perfect dominating set of $G$. The smallest cardinality of a 1-movable restrained perfect dominating set of $G$ is called 1-movable restrained perfect domination number of $G$, denoted by $\gamma_{m r p}^1(G)$. A 1-movable restrained perfect dominating set of $G$ of cardinality $\gamma_{m r p}^1(G)$ is called a $\gamma_{m r p}^1$-set of $G$. This paper characterizes 1 -movable restrained perfect dominating sets in the join and corona of two connected graphs.

Received: December 28, 2022;
Accepted: February 21, 2023;