On total domination subdivision numbers of trees

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number γₜ(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγₜ(G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (Discrete Math. 286 (2004) 195--202) have given a constructive characterization of trees whose total domination subdivision number is~$3$. In this paper, we give new characterizations of trees whose total domination subdivision number is 3.