Perfect matchings in paired domination vertex critical graphs

  • Authors:

  • Shenwei Huang;Erfang Shan;Liying Kang

  • Affiliations:

  • Department of Mathematics, Shanghai University, Shanghai, China 200444;Department of Mathematics, Shanghai University, Shanghai, China 200444;Department of Mathematics, Shanghai University, Shanghai, China 200444

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2012

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Abstract

A vertex subset S of a graph G=(V,E) is a paired dominating set if every vertex of G is adjacent to some vertex in S and the subgraph induced by S contains a perfect matching. The paired domination number of G, denoted by 驴 pr (G), is the minimum cardinality of a paired dominating set of G. A graph with no isolated vertex is called paired domination vertex critical, or briefly 驴 pr -critical, if for any vertex v of G that is not adjacent to any vertex of degree one, 驴 pr (G驴v)驴 pr (G). A 驴 pr -critical graph G is said to be k-驴 pr -critical if 驴 pr (G)=k. In this paper, we firstly show that every 4-驴 pr -critical graph of even order has a perfect matching if it is K 1,5-free and every 4-驴 pr -critical graph of odd order is factor-critical if it is K 1,5-free. Secondly, we show that every 6-驴 pr -critical graph of even order has a perfect matching if it is K 1,4-free.