Bounds on the game isolation number and exact values for paths and cycles

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 NG[X], where NG[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 ιg(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 ιg'(G). In this paper, ιg(Cₙ), ιg(Pₙ), ιg'(Cₙ), and ιg'(Pₙ) are determined for all n. It is proved that there are only two graphs that attain equality in the upper bound ιg(G) ≤ 1/2|V(G)|, and that there are precisely eleven graphs which attain equality in the upper bound ιg'(G) ≤ 1/2|V(G)|. For trees T of order at least three it is proved that ιg(T) ≤ 5/11|V(T)|. A new infinite family of graphs G is also constructed for which ιg(G) = ιg'(G) = 3/7|V(G)| holds.