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Representability of Flag Matroids

Daniel Irving Bernstein, Nathaniel Vaduthala

TL;DR

The paper develops a new cryptomorphic axiom system for flag matroids using feasible sets and studies representability through sequential representations and majors. It proves that representability over a field $\mathbb{K}$ is equivalent to the existence of a $\mathbb{K}$-representable major and is preserved under minors and duals, with a detailed analysis of graphic flag matroids and majors. For full flag matroids, it provides precise forbidden-minor characterizations in the binary and ternary cases: binary matroids are exactly those with no minors isomorphic to $(U_{2,4})$ or $(U_{1,3},U_{2,3})$, while ternary matroids avoid minors $(R)$ or $(R/e,R\setminus e)$ for $R\in\{U_{2,5},U_{3,5},F_7,F^*_7\}$. The results advance understanding of flag matroid representability, link to greedoids, and the interplay between minors, duality, and majors, with some open questions in the $\mathbb{F}_4$ case and in non-graphic settings.

Abstract

We provide a new axiom system for flag matroids, characterize representability of uniform flag matroids, and give forbidden minor characterizations of full flag matroids that are representable over $\mathbb{F}_2$ and $\mathbb{F}_3$.

Representability of Flag Matroids

TL;DR

The paper develops a new cryptomorphic axiom system for flag matroids using feasible sets and studies representability through sequential representations and majors. It proves that representability over a field is equivalent to the existence of a -representable major and is preserved under minors and duals, with a detailed analysis of graphic flag matroids and majors. For full flag matroids, it provides precise forbidden-minor characterizations in the binary and ternary cases: binary matroids are exactly those with no minors isomorphic to or , while ternary matroids avoid minors or for . The results advance understanding of flag matroid representability, link to greedoids, and the interplay between minors, duality, and majors, with some open questions in the case and in non-graphic settings.

Abstract

We provide a new axiom system for flag matroids, characterize representability of uniform flag matroids, and give forbidden minor characterizations of full flag matroids that are representable over and .

Paper Structure

This paper contains 11 sections, 22 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Matroid theory background
  3. Forbidden minors and representability
  4. Flag Matroids
  5. Representable Flag Matroids
  6. Graphic Flag Matroids
  7. Minors and duality
  8. Majors of Flag Matroids
  9. Representability
  10. Binary and Ternary Flag Matroids
  11. Acknowledgments

Key Result

Theorem 2.7

Let $M$ be a matroid. Then $M$ is $\mathbb{F}_2$-representable if and only if $M$ has no minor isomorphic to $U_{2,4}$, and $M$ is $\mathbb{F}_3$-representable if and only if $M$ has no minor isomorphic to $U_{2,5},U_{3,5},F_7$ or $F_7^*$.

Figures (4)

  • Figure 1: Graphic representation of $\mathfrak{F}(K_4, \mathcal{P}_1, \mathcal{P}_2, \mathcal{P}_3, \mathcal{P}_4)$
  • Figure 2: A graph whose matroid is a major of the graphic flag matroid given in Example \ref{['ex: graphic flag matroid']}.
  • Figure 3: From left to right, the graphs $H_1,H_2,G_2,G_3$.
  • Figure 4: From left to right, the graphs $G_3^{bb}$ and $G_3^{rb}$.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Theorem 2.7: oxley
  • Theorem 2.8: oxley
  • Definition 3.1
  • Proposition 3.2: bonin2021delta, oxley
  • ...and 49 more