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 NGX, where NGX 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 ι ₆ (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 ι ₆' (G). In this paper, ι ₆ (Cₙ), ι ₆ (Pₙ), ι ₆' (Cₙ), and ι ₆' (Pₙ) are determined for all n. It is proved that there are only two graphs that attain equality in the upper bound ι ₆ (G) 12|V (G) |, and that there are precisely eleven graphs which attain equality in the upper bound ι ₆' (G) 12|V (G) |. For trees T of order at least three it is proved that ι ₆ (T) 511|V (T) |. A new infinite family of graphs G is also constructed for which ι ₆ (G) = ι ₆' (G) = 37|V (G) | holds.