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

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 $x,y\in S$, $S\mathrm{\backslash }\{x,y\}$ is a dominating set in $G$, or there exist $u,v\in V(G)\mathrm{\backslash }S$ such that $u$ and $v$ are adjacent to $x$ and $y$, respectively, and $(S\mathrm{\backslash }\{x,y\})\cup \{u,v\}$ 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.