On the shape of decomposable trees

A n -vertex graph is said to be decomposable if, for any partition ( λ 1 , … , λ p ) of the integer n , there exists a sequence ( V 1 , … , V p ) of connected vertex-disjoint subgraphs with | V i | = λ i . The aim of the paper is to study the homeomorphism classes of decomposable trees. More precisely, we show that homeomorphism classes containing decomposable trees with an arbitrarily large minimal distance between all pairs of distinct vertices of degree different from 2, is exactly the set of combs.