On a Special Co-cycle Basis of Graphs

• Authors: Telikepalli Kavitha
• Venue: SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
• Year: 2008

In this paper we consider the problems of computing a minimum co-cycle basis and a minimum weakly fundamentalco-cycle basis of a directed graph G. A co-cycle in Gcorresponds to a vertex partition (S,V茂戮驴 S) and a { 茂戮驴 1,0,1} edge incidence vector is associated with each co-cycle. The vector space over 茂戮驴 generated by these vectors is the co-cycle space of G. Alternately, the co-cycle space is the orthogonal complement of the cycle space of G. The minimum co-cycle basis problem asks for a set of co-cycles that span the co-cycle space of Gand whose sum of weights is minimum. Weakly fundamental co-cycle bases are a special class of co-cycle bases, these form a natural superclass of strictly fundamental co-cycle bases and it is known that computing a minimum weight strictly fundamental co-cycle basis is NP-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory-Hu tree of the underlying undirected graph of Gis a minimum co-cycle basis of Gand it is also weakly fundamental.