3-Factor-Criticality in Double Domination Edge Critical Graphs

Abstract

A vertex subset S of a graph G is a double dominating set of G if $$|N[v]\cap S|\ge 2$$|N[v]?S|?2 for each vertex v of G, where N[v] is the set of the vertex v and vertices adjacent to v. The double domination number of G, denoted by $$\gamma _{\times 2}(G)$$?×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if $$\gamma _{\times 2}(G+e)<\gamma _{\times 2}(G)$$?×2(G+e)<?×2(G) for any edge $$e \notin E$$e?E. A double domination edge critical graph G with $$\gamma _{\times 2}(G)=k$$?×2(G)=k is called k-$$\gamma _{\times 2}(G)$$?×2(G)-critical. A graph G is r-factor-critical if $$G-S$$G-S has a perfect matching for each set S of r vertices in G. In this paper we show that G is 3-factor-critical if G is a 3-connected claw-free 4-$$\gamma _{\times 2}(G)$$?×2(G)-critical graph of odd order with minimum degree at least 4 except a family of graphs.