Total isolation game in graphs

Abstract

The total isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $S$ is the set of already played vertices, then a vertex can be selected only if it is adjacent to a vertex that belongs to a (nontrivial) component of the graph $G - N_G(S)$ of order at least $2$ or a vertex that is isolated in $G - N_G(S)$ and belongs to the set $S$, where $N_G(S)$ is the set of vertices adjacent to a vertex in $S$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game total isolation number $\iota_{\rm gt}(G)$ is the number of moves in the Dominator-start game where both players play optimally. We prove that if $G$ is a connected graph of order $n \ge 3$, then $\iota_{\rm gt}(G) < \frac{5}{6}n$. Furthermore if $G$ has minimum degree at least $2$, then we prove that $\iota_{\rm gt}(G) \le \frac{3}{4}n$. More generally, if $G$ is a connected graph of order $n \ge 3$ with minimum degree $\delta$ where $\delta \ge 2$, then we prove that $\iota_{\rm gt}(G) \le \left( \frac{2\delta-1}{3\delta-2} \right) n$. Among other results it is proved that if $G$ is a graph of order $n$ with diameter $2$, then $\iota_{\rm gt}(G) \le \frac{2}{3}n$.