Secure domination critical graphs

A secure dominating set X of a graph G is a dominating set with the property that each vertex u ∈ V G − 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 γ s ( G ) . We are interested in the effect of edge removal on γ s ( G ) , and characterize γ s -ER-critical graphs, i.e. graphs for which γ s ( G − e ) > γ s ( G ) for any edge e of G , bipartite γ s -ER-critical graphs and γ s -ER-critical trees.