Total connected domination game

The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($\gamma_{\rm tcg}(G)$)$\gamma_{\rm cg}(G)$of$G$\mathrm{.}Weshowthat$\gamma_{\rm tcg}(G)\in {\gamma_{\rm cg}(G), \gamma_{\rm cg}(G) + 1, \gamma_{\rm cg}(G) + 2}$, and consequently define $G$ as Class $i$ if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg}

• i$for$i \in {0,1,2}$\mathrm{.}AlargefamilyofClass$0$ graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least $2$. We show that no tree is Class $2$ and characterize Class $1$ trees. We provide an infinite family of Class $2$ bipartite graphs.