The enclaveless competition game

For a subset S of vertices in graph G , vertex v ? is an enclave if and all its neighbors are where neighbor adjacent to . A set enclaveless it does not contain any enclaves. The number ?( ) the maximum cardinality As first observed 1997 by Slater, with n vertices, then ?( + = well-studied domination In this paper, we continue study competition-enclaveless game introduced 2001 Philips Slater defined as follows. Two players take turns constructing maximal one player, Maximizer, tries maximize | Minimizer, minimize ? g ( played when Maximizer starts both play optimally. We among other problems conjecture that isolate-free order ? (1/2) prove for regular graphs claw-free graphs.