Gated independence in graphs

Abstract

If G=(V,E) is a (finite and simple) graph, we call an independent set X a gated independent set in G if for each x∈X, there exists a neighbor y of x such that (X∖{x})∪{y} is an independent set in G. We define the gated independence numbergi(G) of G to be the maximum cardinality of a gated independent set in G. We demonstrate that the gated independence number is closely related to both matching and domination parameters of graphs. We prove that the inequalities im(G)⩽gi(G)⩽mur(G) hold for every graph G, where im(G) and mur(G) denote the induced and uniquely restricted matching numbers of G. On the other hand, we show that γi(G)⩽gi(G) and γpr(G)⩽2gi(G) for every graph G without any isolated vertex, where γi(G) and γpr(G) denote the independence and paired domination numbers. Furthermore, we provide bounds on the gated independence number involving the order, size and maximum degree. In particular, we prove that gi(G)⩾n5 for every n-vertex subcubic graph G without any isolated vertex or any component isomorphic to K3,3, and gi(B)⩽3n8 for every n-vertex connected cubic bipartite graph B.