Total domination subdivision numbers of trees

Abstract 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 γ t ( G ) is the minimum cardinality of a total dominating set of G . The total domination subdivision number sd γ t ( 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. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1 ⩽ sd γ t ( T ) ⩽ 3 . In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.