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

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