Boundary defensive k-alliances in graphs

Abstract

a b s t r a c tWe define a boundary defensive k-alliance in a graph as a set S of vertices with the property that every vertex in S has exactly k more neighbors in S than it has outside of S. In this paper we study mathematical properties of boundary defensive k-alliances. In particular, we obtain several bounds on the cardinality of every boundary defensive k-alliance. Moreover, we consider the case in which the vertex set of a graph can be partitioned into boundary alliances, showing that if a d-regular graph G of order n can be partitioned into two boundary defensive k-alliances X and Y , then |X| = |Y | = n 2 and the algebraic connectivity of G is equal to d − k.