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The excluded minors of the class of spike minors

Sam Bastida, Nick Brettell, Rutger Campbell, George Drummond, Emma Hogan, Charles Semple, Gerry Toft

TL;DR

The paper resolves the open problem of whether the minor-closed class of spikes and their minors has a finite excluded-minor set and explicitly classifies them. By developing a structural theory of spikes, including tipped versus tipless configurations, legs, traversals, and overloaded tipped spikes, the authors perform a rank-based enumeration: the excluded minors for $\mathcal{S}$ consist of uniform-matroid variants and a small finite list of rank-3 examples ($P_7^-$, $P_7^=$) plus $P_8$ as the sole high-rank obstruction; a parallel analysis for 3-connected matroids yields $O_7$, $O_7^-$, $AG(2,3)\setminus e$, $M(\mathcal{W}_4)$, and $\mathcal{W}^4$ among others. The results give a complete minor-certification framework for spike containment and imply binary and ternary corollaries, resolving Mayhew et al.'s question and highlighting the fractal nature of spike-closed classes. Overall, the paper provides a finite, explicit certificate for spike-minor containment with broad implications for the study of minor-closed matroid classes. The high-rank exclusion is uniquely characterized by $P_8$, while the 3-connected regime introduces geometrically flavored obstructions such as $O_7$ and $O_7^-$.

Abstract

Mayhew et al.\ (2021) posed the problem of showing that the minor-closed class of spikes and their minors has a finite set of excluded minors and describing all of them. In this paper, we resolve this problem.

The excluded minors of the class of spike minors

TL;DR

The paper resolves the open problem of whether the minor-closed class of spikes and their minors has a finite excluded-minor set and explicitly classifies them. By developing a structural theory of spikes, including tipped versus tipless configurations, legs, traversals, and overloaded tipped spikes, the authors perform a rank-based enumeration: the excluded minors for consist of uniform-matroid variants and a small finite list of rank-3 examples (, ) plus as the sole high-rank obstruction; a parallel analysis for 3-connected matroids yields , , , , and among others. The results give a complete minor-certification framework for spike containment and imply binary and ternary corollaries, resolving Mayhew et al.'s question and highlighting the fractal nature of spike-closed classes. Overall, the paper provides a finite, explicit certificate for spike-minor containment with broad implications for the study of minor-closed matroid classes. The high-rank exclusion is uniquely characterized by , while the 3-connected regime introduces geometrically flavored obstructions such as and .

Abstract

Mayhew et al.\ (2021) posed the problem of showing that the minor-closed class of spikes and their minors has a finite set of excluded minors and describing all of them. In this paper, we resolve this problem.

Paper Structure

This paper contains 4 sections, 18 theorems, 12 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main1']}
  4. Proof of Theorem \ref{['them: 3con']}

Key Result

Theorem 1.1

A matroid $M$ is a minor of a spike if and only if neither $M$ nor $M^*$ has a minor isomorphic to one of the following matroids:

Figures (5)

  • Figure 1: The rank-$3$ matroids $P_7$, $P_7^-$, and $P_7^=$.
  • Figure 2: $P_8$, the unique excluded minor for $\mathcal{S}$ of rank and corank at least four.
  • Figure 3: The matroids $O_7$ and $O_7^-$.
  • Figure 4: Known circuits of $M/5$, $M/6$, $M/7$, and $M/8$ in the proof of (\ref{['cor:buddy_lemma']}).
  • Figure 5: Known circuits of $M / 4$ and $M / 6$ in the proof of (\ref{['cor:buddy_lemma']}).

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 16 more