Some matching properties in 4- 2-critical graphsI

A vertex subset S of a graph G D .V; E/ is a double dominating set for G if jNTvU \ Sj 2 for each vertex v 2 V, where NTvU D fu juv 2 Eg[fvg. The double domination number of G, denoted by 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 2.GCe/ < 2.G/ for any edge e 62 E. A double domination edge critical graph G with 2.G/ D k is called k- 2.G/-critical. In this paper we first show that G has a perfect matching if G is a connected K1;4-free 4- 2.G/-critical graph of even order 6 except a family of graphs. Secondly, we show that G is bicritical if G is a 2-connected claw-free 4- 2.G/-critical graph of even order with minimum degree at least 3. Finally, we show that G is bicritical if G is a 3-connected K1;4-free 4- 2.G/-critical graph of even order with minimum degree at least 4.