Bounds on a graph's security number

Let G = (V, E) be a graph. A set S subset of V is a defensive alliance if vertical bar N[x] boolean AND S vertical bar >= vertical bar N [x] - S vertical bar for every X is an element of S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X subset of S, not just singletons, can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. The security number s(G) of G is the cardinality of a smallest secure set. Bounds on s(G) are presented. Published by Elsevier B.V.