SOME CHARACTERIZATIONS OF 1-MOVABLE RESTRAINED CONNECTED DOMINATING SETS IN THE JOIN OF GRAPHS

Abstract

Let $G$ be a connected graph. A set $S \subseteq V(G)$ is a 1-movable restrained connected dominating set in $G$ if $S=V(G)$ or $S$ is a restrained connected dominating set of $G$ and for every $v \in S$, either $S \backslash\{v\}$ is a restrained connected dominating set or there exists a vertex $u \in(V(G) \backslash S) \cap N_G(v)$ such that $(S \backslash\{v\}) \bigcup\{u\}$ is a restrained connected dominating set of $G$. The 1-movable restrained connected domination number of $G$, denoted by $\gamma_{m r c}^1(G)$, is the smallest cardinality of a 1-movable restrained connected dominating set of $G$. A 1-movable restrained connected dominating set with cardinality $\gamma_{m r c}^1(G)$ is called a $\gamma_{m r c}^1$-set in $G$. This paper characterizes 1movable restrained connected dominating set in the join of two graphs.