The enclaveless competition game

Abstract

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