Domination subdivision numbers of trees

A set SS of vertices of a graph G=(V,E)G=(V,E) is a dominating set   if every vertex of V(G)∖SV(G)∖S is adjacent to some vertex in SS. The domination number   γ(G)γ(G) is the minimum cardinality of a dominating set of GG. The domination subdivision number   sdγ(G) is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree TT of order at least 3, 1≤sdγ(T)≤3. In this paper, we give two characterizations of trees whose domination subdivision number is 3 and a linear algorithm for recognizing them.