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 $|N[v]\cap S|\ge 2$ 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 $\gamma _{\times 2}(G)$, is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if $\gamma _{\times 2}(G+e)<\gamma _{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph G with $\gamma _{\times 2}(G)=k$ is called k-$\gamma _{\times 2}(G)$-critical. A graph G is r-factor-critical if $G-S$ 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-$\gamma _{\times 2}(G)$-critical graph of odd order with minimum degree at least 4 except a family of graphs.