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Small separations in pinch-graphic matroids

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Abstract

Even-cycle matroids are elementary lifts of graphic matroids. Pinch-graphic matroids are even-cycle matroids that are also elementary projections of graphic matroids. In this paper we analyze the structure of 1-, 2-, and 3-separations in these matroids. As a corollary we obtain a polynomial-time algorithm that reduces the problem of recognizing pinch-graphic matroids to internally 4-connected matroids. Combining this with earlier results (Guenin and Heo in Recognizing even-cycle and even-cut matroids manuscript, 2020; Guenin and Heo in Recognizing pinch-graphic matroids manuscript, 2020) we obtain a polynomial-time algorithm for recognizing even-cycle matroids and we obtain a polynomial-time algorithm for recognizing even-cut matroids.

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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 to the referees for their work and many discerning comments that improved the paper.

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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, South 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. Small separations in pinch-graphic matroids. Math. Program. 204, 81–111 (2024). https://doi.org/10.1007/s10107-023-01950-8

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