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Recognizing even-cycle and even-cut matroids

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Abstract

Even-cycle matroids are elementary lifts of graphic matroids and even-cut matroids are elementary lifts of cographic matroids. We present a polynomial algorithm to check if a binary matroid is an even-cycle matroid and we present a polynomial algorithm to check if a binary matroid is an even-cut matroid. These two algorithms rely on a polynomial algorithm (to be described in a pair of follow-up papers) to check if a binary matroid is pinch-graphic.

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Notes

  1. \([n]=\{1,\ldots ,n\}\).

  2. \(\delta _G(v)\) denotes the set of non-loop edges of graph G incident to v.

  3. For a set A and element of the ground set a we write \(A\cup a\) for \(A\cup \{a\}\) and write \(A-a\) for \(A-\{a\}\).

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Acknowledgements

We would like to thank Irene Pivotto for numerous insightful and lively discussions on the subject of even-cycle and even-cut matroids and on the problem of designing recognition algorithms. We are grateful for the work of the referees that improved the quality of the paper.

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Authors and Affiliations

  1. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Canada

    Bertrand Guenin

  2. Institute for Advanced Study (KIAS), 85 Hoegi-ro, Dongdaemun-gu, Seoul, South Korea

    Cheolwon Heo

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  1. Bertrand Guenin
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  2. Cheolwon Heo
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Correspondence to Bertrand Guenin.

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Guenin, B., Heo, C. Recognizing even-cycle and even-cut matroids. Math. Program. 202, 515–542 (2023). https://doi.org/10.1007/s10107-023-01944-6

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