Abstract
The total domination number γt(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In this paper we prove two new upper bounds on the total domination number of a tree related to particular support vertices (vertices adjacent to leaves) of the tree. One of these bounds improves a 2004 result of Chellali and Haynes [1]. In addition, we prove some bounds on the total domination ratio of trees.
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