On cycle cones and polyhedra

Abstract

Given an undirected graph G and a cost associated with each edge, the weighted girth problem is to find a simple cycle of G having minimum total cost. We consider several variants of the weighted girth problem, some of which are NP-hard and some of which are solvable in polynomial time. We also consider the polyhedra associated with each of these problems. Two of these polyhedra are the cycle cone of G, which is the cone generated by the incidence vectors of cycles of G, and the cycle polytope of G, which is the convex hull of the incidence vectors of cycles of G. First we give a short proof of Seymour's characterization of the cycle cone of G. Next we give a polyhedral composition result for the cycle polytope of G. In particular, we prove that if G decomposes via a 3-edge cut into graphs G 1 and G 2, say, then defining linear systems for the cycle polytopes of G 1 and G 2 can be combined in a certain way to obtain a defining linear system for the cycle polytope of G. We also describe a polynomial decomposition-based algorithm for the weighted girth problem on Halin graphs, and we give a complete linear description for the cycle polytope of G, in the case G is a Halin graph.