The Metric Cutpoint Partition Problem

Abstract

Let G?=?(V, E,w) be a graph with vertex and edge sets V and E, respectively, and w: E ? $ \mathbb{R}^{+} $ a function which assigns a positive weight or length to each edge of G. G is called a realization of a finite metric space (M, d), with M?=?{1, ..., n} if and only if {1, ..., n} ? V and d(i, j) is equal to the length of the shortest chain linking i and j in G ?i, j?=?1, ..., n. A realization G of (M, d), is called optimal if the sum of its weights is minimal among all the realizations of (M, d). A cutpoint in a graph G is a vertex whose removal strictly increases the number of connected components of G. The Metric Cutpoint Partition Problem is to determine if a finite metric space (M, d) has an optimal realization containing a cutpoint. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M, d) from optimal realizations of subspaces that do not contain any cutpoint.