Trees whose domination subdivision number is one

A set S of vertices of a graphG = (V,E) is a dominating set if every vertex of V (G)\S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Velammal in his Ph.D. thesis [Manonmaniam Sundaranar University, Tirunelveli, 1997] showed that for any tree T of order at least 3, 1 ≤ sdγ(T ) ≤ 3. Furthermore, Aram, Favaron and Sheikholeslami, recently, in their paper entitled “Trees with domination subdivision number three,” gave two characterizations of trees whose domination subdivision number is three. In this paper we characterize all trees whose domination subdivision number is one.