Double domination edge removal critical graphs

In a graph, a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number γ×2(G) is the minimum cardinality of a double dominating set of G. A graph G without isolated vertices is called edge removal critical with respect to double domination, or just γ×2-critical, if the removal of any edge increases the double domination number. We first give a necessary and sufficient condition for γ×2-critical graphs. Then we give a characterization of γ×2-critical graphs for some classes of graphs including trees, P4-free and P5-free graphs. Finally, we investigate γ×2-critical graphs having double domination number 3 or 4.