Z-domination game

The Z-domination game is a variant of the domination game in which each newly selected vertex $u$ in the game must have a not yet dominated neighbor, but after the move all vertices from the closed neighborhood of $u$ are declared to be dominated. The Z-domination game is the fastest among the five natural domination games. The corresponding game Z-domination number of a graph $G$ is denoted by ${\gamma }_{Zg}(G)$. It is proved that the game domination number and the game total domination number of a graph can be expressed as the game Z-domination number of appropriate lexicographic products. Graphs with a Z-insensitive property are introduced and it is proved that if $G$ is Z-insensitive, then ${\gamma }_{Zg}(G)$ is equal to the game domination number of $G$. Weakly claw-free graphs are defined and proved to be Z-insensitive. As a consequence, ${\gamma }_{Zg}(P_n)$ is determined, thus sharpening an earlier related approximate result. It is proved that if ${\gamma }_{Zg}(G)$ is an even number, then ${\gamma }_{Zg}(G)$ is strictly smaller than the game L-domination number. On the other hand, families of graphs are constructed for which all five game domination numbers coincide. Graphs $G$ with $\gamma_{Zg}(G) = \gamma(G)$ are also considered and computational results which compare the studied invariants in the class of trees on at most $16$ vertices reported.