On a graph\u27s security number

A secure set S subset of V of graph G = (V. E) is a set whose every nonempty subset can be successfully defended from an attack, under appropriate definitions of attack and defended. The set S is secure when vertical bar N[X]boolean AND S vertical bar \u3e = vertical bar N[X] - S vertical bar for every X subset of S. The smallest cardinality of a secure set in G is the security number of G. New bounds for the security number are established, the effect of some graph modifications on the security number is investigated, and the exact value of the security number for some families of graphs is given. (C) 2009 Elsevier B.V. All rights reserved