Matching Properties in Total Domination Vertex Critical Graphs |
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Authors: | Haichao Wang Liying Kang Erfang Shan |
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Institution: | 1. Department of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, People’s Republic of China 2. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China
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Abstract: | A vertex subset S of a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ t (G?v) < γ t (G). A total domination vertex critical graph G is called k-γ t -critical if γ t (G) = k. In this paper we first show that every 3-γ t -critical graph G of even order has a perfect matching if it is K 1,5-free. Secondly, we show that every 3-γ t -critical graph G of odd order is factor-critical if it is K 1,5-free. Finally, we show that G has a perfect matching if G is a K 1,4-free 4-γ t (G)-critical graph of even order and G is factor-critical if G is a K 1,4-free 4-γ t (G)-critical graph of odd order. |
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