ELSEVIER Operations Research Letters 19 (1996) 71-78 k-edge connected polyhedra on series-parallel graphs M. Didi Biha, A.R. Mahjoub * Laboratoire SPO, Dbpartementd'lnformatique, Universit~ de Bretagne Occidentale, 6 avenue VictorLe Gorgeu, B.P. 809, 29285 Brest Cedex, France Received 1 August 1994; revised 1 January 1996 Abstract We give a complete description of the k-edge connected spanning subgraph polytope (for all k) on series-parallel graphs. Keywords: k-edge connected subgraphs; Polytopes; Series-parallel graphs 1. Introduction The graphs we consider are finite, undirected, loop- less and may have multiple edges. A graph is denoted by G=(V,E) where V is the set of nodes and E is the set of edges of G. A graph G = (V,E) is called k-edge connected (where k is a positive integer) if for any pair of nodes i, j E V, there are at least k edge-disjoint paths from itoj. Given a graph G =(V,E) and a weight function co on E that associates with an edge e E E, the weight co(e) E R, the k-edoe connected spanning subgraph problem (kECSP for short) is to find a k-edge con- nected subgraph H = (V,F) of G, spanning all the nodes in V, such that ~ eeFco(e) is minimum. The kECSP arises in the design of communication and transportation networks [4, 7,26,27]. It is NP-hard for k ~> 2. For k = 1, the problem reduces to the min- imum spanning tree problem and thus can be solved in polynomial time. * Corresponding author. If G = (V,E) is a graph and F C_E, the incidence vector ofF will be denoted by X F. The convex hull of the incidence vectors of all edge sets of k-edge con- nected spanning subgraphs of G is called the k-edge connected spanning subgraph polytope and denoted by kECP(G), that is, kECP(G) = conv{x F E ~eI(V,F) is k-edge connected spanning subgraph of G}. In this paper we discuss the polytope kECP(G), we give a complete description of this polytope on series- parallel graphs. The kECSP and the polytope kECP(G) have received special attention in the past few years. Grftschel and Monma [19] and Gr6tschel et al. [20-22] studied the kECSP within the framework of a more general model related to the design of mini- mum cost survivable networks. In [19] basic facets of kECP(G) are described. In [20-22], further classes of valid inequalities and facets are characterized, and a cutting plane algorithm along with computational re- sults are discussed. A complete survey of that model can be found in [27]. 0167-6377/96/$15.00 Copyright(~) 1996 ElsevierScienceB.V, All rights reserved PI! SO167-6377(96)00015-6