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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 of G, denoted ${\gamma }_{\times 2}\left(G\right)$, is the minimum cardinality among all double dominating sets of G. We consider the effects of vertex removal on the double domination number of a graph. A graph G is ${\gamma }_{\times 2}$-vertex critical graph (${\gamma }_{\times 2}$-vertex stable graph, respectively) if the removal of any vertex different from a support...