Formula omitted]-distance secure sets in graphs

Abstract

Let G=V,E be a graph and S be a non empty subset of V. For any s∈S, a vertex of Ns-S is an attacker of s and a vertex of Ns∩S is a defender of s. An attack on S is a collection of mutually disjoint sets of attackers of vertices in S. A defense of S is a collection of mutually disjoint sets of defenders of vertices in S. The set S is a secure set if for every attack on S, there is a defense of S such that the defending set has at least as many vertices as in the attacking set corresponding to every vertex of S. In this paper, k-distance secure sets are defined as a generalization secure sets. A characterization of k-distance secure sets is obtained. Further k-distance security numbers of certain classes of graphs are determined.