Decomposable trees: a polynomial algorithm for tripodes

In this article, we deal with graphs modelling interconnection networks of parallel systems (parallel computers, networks of workstations, etc.). We want to share the nodes of such a network between many users, each one needing a given number of nodes. Thus, a graph G with N vertices is said to be decomposable if for each set { n 1,…, n k } whose sum is equal to N, there exists a partition V 1,…, V k of V( G) such that for each i, 1⩽ i⩽ k, | V i |= n i and the subgraph induced by V i is connected. We show that determining whether a given tripode (three disjoint chains connected by one extremity to a same new vertex) is decomposable can be done by a polynomial algorithm.