Domination subdivision numbers of trees

A set S of vertices of a graph G = ( 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  sd γ ( G ) is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree T of order at least 3, 1 ≤ sd γ ( T ) ≤ 3 . In this paper, we give two characterizations of trees whose domination subdivision number is 3 and a linear algorithm for recognizing them.