Union of all the Minimum Cycle Bases of a Graph

Source abstract

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 ${\cal C_R}$. A relevant cycle can also be defined 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 ${\cal C_R}$ in $O(\nu m^3)$ (where $\nu$ 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 $O(n\,|{\cal C_R}|)$. A polynomial method is also given that computes the number of relevant cycles without listing all of them.