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

Let $G$ be a connected graph. A non-empty $T\subseteq V(G)$ is a $2$-$\mathit{\text{movable}}\mathit{\text{total}}\mathit{\text{dominating}}\mathit{\text{set}}$ of $G$ if $T$ is a total dominating set and for every pair $x,y\in T$, $T\mathrm{\backslash }\{x,y\}$ is a total dominating set in $G$, or there exist $u,v\in V(G)\mathrm{\backslash }T$ such that $u$ and $v$ are adjacent to $x$ and $y$, respectively, and $(T\mathrm{\backslash }\{x,y\})\cup \{u,v\}$ is a total dominating set in $G$. The $2$-$\mathit{\text{movable}}\mathit{\text{total}}\mathit{\text{domination}}\mathit{\text{number}}$ 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.