Minimum Cycle Bases in Graphs Algorithms and Applications

Kurt Mehlhorn

Abstract: A cycle basis of a graph is a family of cycles which spans all cycles of the graph. In an undirected graph, a cycle is simply a set of edges with respect to which every vertex has even degree. We view cycles as vectors indexed by edges. The entry for an edge is one if the edge belongs to the cycle and is zero otherwise. Addition of cycles corresponds to vector addition modulo 2 (symmetric difference of the underlying edge sets). In this way, the cycles of a graph form a vector space and a cycle basis is simply a basis of this vector space. The notion for directed graphs is slightly more involved.