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Recognizing pinch-graphic matroids

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Abstract

Even-cycle matroids are elementary lifts of graphic matroids. An even-cycle matroid is pinch-graphic if it has a signed-graph representation with a blocking pair. We present a polynomial algorithm to check if an internally 4-connected binary matroid is pinch-graphic. Combining this with a result in Guenin and Heo (Small separations in pinch-graphic matroids. Math Program (2023). https://doi.org/10.1007/s10107-023-01950-8) this allows us to check, in polynomial time, if an arbitrary binary matroid is pinch-graphic.

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Notes

  1. For sets A, B we denote by \(A-B\) the set \(\{a\in A:a\notin B\}\).

  2. For a graph H and vertex v, \(\delta _H(v)\) denotes the set of non-loop edges incident to v.

  3. 3 choices for which pairs of terminals get identified, and \(2\times 2\) choices for the signature.

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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 careful work of the referees and the many insightful suggestions.

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Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, N2L3G1, Canada

    Bertrand Guenin

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

    Cheolwon Heo

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

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Supported by NSERC grant 238811 and ONR grant N00014-12-1-0049. An extended abstract of these results appeared in Heo, C., Guenin, B. Recognizing Even-Cycle and Even-Cut Matroids, 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings, 182–195 (2020).

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Guenin, B., Heo, C. Recognizing pinch-graphic matroids. Math. Program. 204, 113–134 (2024). https://doi.org/10.1007/s10107-023-01951-7

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  • DOI: https://doi.org/10.1007/s10107-023-01951-7

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