Decomposing trees with large diameter

An n-vertex graph is said to be decomposable for a partition (λ1,…,λp) of the integer n if there exists a sequence (V1,…,Vp) of connected vertex-disjoint subgraphs with |Vi|=λi. An n-vertex graph is said to be decomposable if this graph is decomposable for all the partitions of the integer n. We are interested in decomposable trees with large diameter. We show that any n-vertex tree T with diameter n−α is decomposable for all the partitions of n which contain at least α distinct integers. This structural result provides an algorithm to decide if an n-vertex tree T with diameter n−α is decomposable in time nO(α).