Secure domination critical graphs

A Secure dominating set X of a graph G is a dominating set with the property that each vertex u is an element of V-G - X is adjacent to a vertex nu is an element of X such that (X - {nu}) U {u} is dominating. The minimum cardinality of such a set is called the secure domination number, denoted by gamma(s)(G). We are interested in the effect of edge removal on gamma(s)(G), and characterize gamma(s) ER-critical graphs, i.e. graphs for which gamma(s)(G - e) > gamma(s)(G) for any edge e of G, bipartite gamma(s)-ER-critical graphs and gamma(s)-ER-critical trees. (C) 2008 Elsevier B.V. All rights reserved.