Some results on matching and total domination in graphs

Let G be a graph. A set S of vertices of G is called a total dominating set of G if every vertex of G is adjacent to at least one vertex in S. The total domination number γt(G) and the matching number α'(G) of G are the cardinalities of the minimum total dominating set and the maximum matching of G, respectively. In this paper, we introduce an upper bound of the difference between γt(G) and α'(G). We also characterize every tree T with γt(T) ≤ α'(T), and give a family of graphs with γt(G) ≤ α'(G).