Matching Properties in Total Domination Vertex Critical Graphs

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 γt(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, γt(G−v) < γt(G). A total domination vertex critical graph G is called k-γt-critical if γt(G) = k. In this paper we first show that every 3-γt-critical graph G of even order has a perfect matching if it is K1,5-free. Secondly, we show that every 3-γt-critical graph G of odd order is factor-critical if it is K1,5-free. Finally, we show that G has a perfect matching if G is a K1,4-free 4-γt(G)-critical graph of even order and G is factor-critical if G is a K1,4-free 4-γt(G)-critical graph of odd order.