strong edge criticality for cardinality-redundance in graphs

A vertex v in a graph g is said to be over-dominated by a subset s of vertices of g if |n[v] ∩ s| ≥ 2. the cardinality–redundance of s, which we denote by cr(s), is the number of vertices of g that are over-dominated by s. the cardinality–redundance number of a graph g, which we denote by cr(g), is the minimum among all cardinality–redundances cr(s) taken over all dominating sets s. in this paper, we study those graphs whose cardinality–redundance decreases by two upon the removal of any arbitrary edge. we refer to such graphs as cardinality–redundance strong edge critical graphs. we give a general characterization for all cardinality–redundance strong edge critical graphs, and then focus on the cardinality–redundance strong edge critical graphs having cardinality–redundance 2, and present several characterizations for these graphs.