Union of all the Minimum Cycle Bases of a Graph

The perception of cyclic structures is a crucial step in the analysis of graphs. To describe the cycle vector space of a graph, a minimum cycle basis can be computed in polynomial time using an algorithm of Horton, 1987]. But the set of cycles corresponding to a minimum basis is not always relevant for analyzing the cyclic structure of a graph. This restriction is due to the fact that a minimum cycle basis is generally not unique for a given graph. Therefore, the smallest canonical set of cycles which describes the cyclic structure of a graph is the union of all the minimum cycle bases. This set of cycles is called the set of relevant cycles and denoted by C R. A relevant cycle can also be deened as a cycle which is not the sum of shorter cycles. A polynomial algorithm is presented that computes a compact representation of the potentially exponential-sized set C R in O(m 3) (where denotes the cyclomatic number). This compact representation consists of a polynomial number of relevant cycle prototypes from which all the relevant cycles can be listed in O(n jC R j). A polynomial method is also given that computes the number of relevant cycles without listing all of them.