Computation of Cycle Bases in Surface Embedded Graphs

Abstract

We study the problem of finding a cycle basis, a minimum weight set of independent cycles that form a basis of the cycle space for a given graph. We focus on finding the minimum cycle basis of directed graphs. This is a more complicated problem compared to the undirected case as the underlying field is Q for directed graphs instead of Z2 for undirected, which causes problems in the speed of calculations. Previously the fastest known deterministic algorithm to find the minimum cycle basis of a directed graph runs in O(m3n + m2n 2 log n) time [11]. We concentrate on graphs embedded on a surface of genus g. We modify algorithms for undirected graphs to work on directed graphs. We present an O(mn2 g 2 log g + mω+1) time algorithm to find the minimum cycle basis of a directed graph embedded on a surface of genus g. We also give an improvement on the minimum cycle basis in the undirected case