The Metric Cutpoint Partition Problem

Let G = (V,E,w) be a graph with vertex and edge sets V and E, respectively, and w : E → IR 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 said 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 ⋆ Supported by grant PA002-104974/2 from the Swiss National Science Foundation. 2 Alain Hertz, Sacha Varone an algorithm that constructs an optimal realization of (M,d) from optimal realizations of subspaces that do not contain any cutpoint.