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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机译：如果G的每个顶点都与S中的某个顶点相邻，则图G =（V，E）的顶点子集S是总控制集。G的总控制数由γ t 表示（G）是G的总支配集合的最小基数。如果对于G的任何顶点v不与一阶顶点γ相邻，则没有孤立顶点的图G被认为是总支配顶点关键。 t （Gv）<γ t （G）。如果γ t （G）= k，则总支配顶点临界图G称为k-γ t -critical。在本文中，我们首先证明，如果没有K 1,5 ，则每个偶数阶的3-γ t -临界图G都有一个完美的匹配。其次，我们表明，如果每个奇数阶的3-γ t 临界图G无K 1,5 ，则它是因子临界的。最后，我们证明，如果G是偶数阶的无K 1,4 的4-γ t （G）临界图，并且G为如果G是奇数阶的无K 1,4 的4-γ t （G）临界图，则该因子为临界。
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