Some matching properties in 4-γ×2-critical graphs

Abstract

A vertex subset S of a graph G = ( V , E ) is a double dominating set for G if | N [ v ] ∩ S | ≥ 2 for each vertex v ∈ V , where N [ v ] = { u | u v ∈ E } ∪ { v } . The double domination number of G , denoted by γ × 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 γ × 2 ( G + e ) < γ × 2 ( G ) for any edge e ∉ E . A double domination edge critical graph G with γ × 2 ( G ) = k is called k - γ × 2 ( G ) - critical . In this paper we first show that G has a perfect matching if G is a connected K 1 , 4 -free 4- γ × 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- γ × 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 K 1 , 4 -free 4- γ × 2 ( G ) -critical graph of even order with minimum degree at least 4.