N-Critical Matroids

Abstract

Let M and N be 3-connected matroids; we say that M is N-critical if M has an N-minor, but for each \(x\in E(M)\), \(M\backslash x\) is not 3-connected or \(M\backslash x\) has no N-minor. We establish that if M is an N-critical matroid with \(r^*(M)>\max \{3,r^*(N)\}\), then M has an element x such that either \(\mathrm{co}(M\backslash x)\) is N-critical or M has a coline \(L^*\) with \(|L^*|\ge 3\) such that \(M\backslash L^*\) is N-critical. As a corollary we get a chain theorem for the class of minimally 3-connected matroids. This chain theorem generalizes a previous one of Anderson and Wu for binary matroids.

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Funding

The second author is partially supported by PSC-CUNY Grant number #62231-00 50.

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Correspondence to J. P. Costalonga.

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Costalonga, J.P., Kingan, S.R. N-Critical Matroids. Graphs and Combinatorics (2021). https://doi.org/10.1007/s00373-021-02281-1

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Keywords

  • Splitter theorem
  • Chain theorem
  • Matroid connectivity
  • Matroid minors
  • Minimal 3-connected

Mathematics Subject Classification

  • MSC, 05B35