Bounds on a graphʹs security number Original Research Article

Let image be a graph. A set image is a defensive alliance if image for every image. 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 image, 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 image of G is the cardinality of a smallest secure set. Bounds on image are presented.