The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $\iota_{ \rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $\iota_{ \rm g}'(G)$. In this paper, $\iota_{ \rm g}(C_n)$, $\iota_{ \rm g}(P_n)$, $\iota_{ \rm g}'(C_n)$, and $\iota_{ \rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $\iota_{ \rm g}(G) \le \frac{ 1}{ 2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $\iota_{ \rm g}'(G) \le \frac{ 1}{ 2}|V(G)|$. For trees $T$ of order at least three it is proved that $\iota_{ \rm g}(T) \le \frac{ 5}{ 11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $\iota_{ \rm g}(G) = \iota_{ \rm g}'(G) = \frac{ 3}{ 7}|V(G)|$ holds.
Categories: math.CO