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Journal Articles The Electronic Journal of Combinatorics Year : 1997

Union of all the Minimum Cycle Bases of a Graph

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.
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Dates and versions

hal-04478606 , version 1 (27-02-2024)

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Philippe Vismara. Union of all the Minimum Cycle Bases of a Graph. The Electronic Journal of Combinatorics, 1997, 4 (1), ⟨10.37236/1294⟩. ⟨hal-04478606⟩
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