A new lower bound on the total domination number of a tree

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The order-sum number orda(G) is the smallest integer k such that the sum of the first k terms of the non-increasing degree sequence of G is at least as large as the order of G. In this paper, we introduce the concept of the order-sum number, use it to establish a lower bound on the total domination number of a tree and show that this lower bound is an improvement to a previous result of Chellali and Haynes.