On 2-Movable Domination in the Join and Corona of Graphs

Abstract

Let $G$ be a connected graph. A non-empty $S\subseteq V(G)$ is a $2$-movable dominating set of $G$ if $S$ is a dominating set and for every pair $x,y \in S$, $S\backslash \{x, y\}$ is a dominating set in $G$, or there exist $u, v \in V(G) \backslash S$ such that $u$ and $v$ are adjacent to $x$ and $y$, respectively, and $(S \backslash \{x,y\}) \cup \{u,v\}$ is a dominating set in $G$. The $2$-movable domination number of $G$, denoted by $\gamma_{m}^{2}(G)$, is the minimum cardinality of a 2-movable dominating set of $G$. A 2-movable dominating set with cardinality equal to $\gamma_{m}^{2}(G)$ is called $\gamma_{m}^{2}$-set of $G$. This paper present the 2-movable domination number in the corona and join of graphs.