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 Mobius ladder $$\mathrm{ML}_{2k}$$ is 2k- $$\gamma _{\mathrm{tg}}$$ -critical.