Secure domination critical graphs

AbstractA secure dominating set X of a graph G is a dominating set with the property that each vertex u∈VG−X is adjacent to a vertex v∈X such that (X−{v})∪{u} is dominating. The minimum cardinality of such a set is called the secure domination number, denoted by γs(G). We are interested in the effect of edge removal on γs(G), and characterize γs-ER-critical graphs, i.e. graphs for which γs(G−e)>γs(G) for any edge e of G, bipartite γs-ER-critical graphs and γs-ER-critical trees