Minimum Path Bases

Abstract

In this paper we introduce a new vector space associated with the paths and cycles in a graph (this space properly includes the well-known cycle space). We present a polynomial algorithm for finding a minimum weight basis for this space; we then present an application of this algorithm. The application is an algorithm that finds in a graph a minimum weight subgraph whose cycle space has specified dimension k (when k = 1, this is the problem of finding a minimum weight cycle in a graph). We then show how this algorithm easily provides a new algorithm for the planar k-split problem; for a planar graph, this is the problem of finding a minimum weight set of edges whose deletion results in a graph with k components (when k = 2, this is the problem of finding a minimum cut in a planar graph). The algorithm for the application is polynomial for fixed k. In particular, for the planar three-split problem, our algorithm can be implemented with worst case time complexity O(| V| 3), whereas the best previous algorithm has complexity O(| V| 6 log| V|).