The isolation game for regular graphs

Robinson, A.G. and A.J. Goldman, The isolation game for regular graphs, Discrete Mathematics 112 (1993) 173-184. In Ringeisen’s Isolation Game on a graph, two players alternately ‘switch’ at successive vertices v not previously switched: the switching operation deletes all edges incident with v, and creates new edges between t’ and those vertices not previously adjacent to it. The game is won when a vertex is first isolated. A previous paper established that (with best play) such games can be won only either very early or very late, implying that most graphs are nonwinnable by either player. Here we focus on regular graphs, showing that their Isolation Games cannot be won unless they can be won extremely early, and identifying the winnable regular graphs explicitly.