Secure domination critical graphs

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