A degree bound on decomposable trees

A n  -vertex graph is said to be decomposable if for any partition (λ1,…,λp)(λ1,…,λp) of the integer n  , there exists a sequence (V1,…,Vp)(V1,…,Vp) of connected vertex-disjoint subgraphs with |Vi|=λi|Vi|=λi. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polynomial time algorithm to decide if a multipode (a tree with only one vertex of degree greater than 2) is decomposable. We also exhibit two families of decomposable trees: arbitrary large trees with one vertex of degree 4, and trees with an arbitrary number of degree-3 vertices.