Infinite Families of Circular and Möbius Ladders that are Total Domination Game Critical

Abstract

Let $\gamma _\mathrm{tg}(G)$ denote the game total domination number of a graph G, and let G|v mean that a vertex v of G is declared to be already totally dominated. A graph G is total domination game critical if $\gamma _\mathrm{tg}(G|v) < \gamma _\mathrm{tg}(G)$ holds for every vertex v in G. If $\gamma _\mathrm{tg}(G) = k$, then G is further called k-$\gamma _\mathrm{tg}$-critical. In this paper, we prove that the circular ladder $C_{4k} \,\square \,K_2$ is 4k-$\gamma _{\mathrm{tg}}$-critical and that the Möbius ladder $\mathrm{ML}_{2k}$ is 2k-$\gamma _{\mathrm{tg}}$-critical.