Isolation Game on Graphs

Given a graph G and a family of graphs F F , an F F -isolating set, as introduced by Caro and Hansberg, is any set S⊂ V(G) S ⊂ V ( G ) such that $$G - N[S]$$ G - N [ S ] contains no member of F F as a subgraph. In this paper, we introduce a game in which two players with opposite goals are together building an F F -isolating set in G . Following the domination games, Dominator (Staller) wants that the resulting F F -isolating set obtained at the end of the game, is as small (as big) as possible, which leads to the graph invariant called the game F F -isolation number, denoted ι g(G, F) ι g ( G , F ) . We prove that the Continuation Principle holds in the F F -isolation game, and that the difference between the game F F -isolation numbers when either Dominator or Staller starts the game is at most 1. Considering two arbitrary families of graphs F F and F' F ′ , we find relations between them that ensure ι g(G,F') ≤ ι g(G,F) ι g ( G , F ′ ) ≤ ι g ( G , F ) for any graph G . A special focus is given on the isolation game, which takes place when F=₂\ <mml:math xmlns:mml="http