Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Describing Quasi-Graphic Matroids

Nathan Bowler, Daryl Funk, Daniel Slilaty

TL;DR

The paper resolves a corrected, complete description of quasi-graphic matroids by enforcing a connectivity condition in the main theorem and develops a comprehensive framework using bracelet functions and proper tripartitions to characterize these matroids via graphs. It introduces and analyzes intermediate and biased-graphic matroids, showing how quasi-graphic matroids sit between frame and lifted-graphic matroids and are captured by both bracelet-function and tripartition formalisms, with minor-closure and 2-sum decompositions clarifying structure. A central contribution is the equivalence between having a proper bracelet function or a proper tripartition and being quasi-graphic, under suitable connectivity assumptions; this extends to biased-graphic matroids and yields a minor-closed, richly described class that includes broken handcuff constructions. The results have implications for representability, minor theory, and efficient checking of quasi-graphic axioms, situating quasi-graphic matroids as a robust generalization of frame and lifted-graphic matroids with a detailed graphical/biased-graph framework.

Abstract

This is a revised version of our original paper (arXiv:1808.00489v2) incorporating the corrections published in a corrigendum (arXiv:1808.00489v3). Our main theorem as originally stated was missing the required assumption that matroids should be connected. Those unfamiliar with the original paper will find in this version a complete, correct description of quasi-graphic matroids, sparing them the inconvenience of having to read both the original paper and a separate corrigendum. We also present here some new results that do not appear in our original paper nor its corrigendum. These appear in Section 6. Of particular interest to readers familiar with the original paper and its corrigendum may be the following result. Given a matroid and a graph, of the four axioms for quasi-graphic matroids, three may be checked in time polynomial in the size of the ground set, but the fourth axiom in general cannot. It is desirable to have such a check that could be carried out in polynomial time. We provide such an alternative (Theorem 6.20).

Describing Quasi-Graphic Matroids

TL;DR

The paper resolves a corrected, complete description of quasi-graphic matroids by enforcing a connectivity condition in the main theorem and develops a comprehensive framework using bracelet functions and proper tripartitions to characterize these matroids via graphs. It introduces and analyzes intermediate and biased-graphic matroids, showing how quasi-graphic matroids sit between frame and lifted-graphic matroids and are captured by both bracelet-function and tripartition formalisms, with minor-closure and 2-sum decompositions clarifying structure. A central contribution is the equivalence between having a proper bracelet function or a proper tripartition and being quasi-graphic, under suitable connectivity assumptions; this extends to biased-graphic matroids and yields a minor-closed, richly described class that includes broken handcuff constructions. The results have implications for representability, minor theory, and efficient checking of quasi-graphic axioms, situating quasi-graphic matroids as a robust generalization of frame and lifted-graphic matroids with a detailed graphical/biased-graph framework.

Abstract

This is a revised version of our original paper (arXiv:1808.00489v2) incorporating the corrections published in a corrigendum (arXiv:1808.00489v3). Our main theorem as originally stated was missing the required assumption that matroids should be connected. Those unfamiliar with the original paper will find in this version a complete, correct description of quasi-graphic matroids, sparing them the inconvenience of having to read both the original paper and a separate corrigendum. We also present here some new results that do not appear in our original paper nor its corrigendum. These appear in Section 6. Of particular interest to readers familiar with the original paper and its corrigendum may be the following result. Given a matroid and a graph, of the four axioms for quasi-graphic matroids, three may be checked in time polynomial in the size of the ground set, but the fourth axiom in general cannot. It is desirable to have such a check that could be carried out in polynomial time. We provide such an alternative (Theorem 6.20).

Paper Structure

This paper contains 24 sections, 47 theorems, 30 equations, 5 figures.

Table of Contents

  1. Context and motivation
  2. Frame matroids
  3. Lifted-graphic matroids
  4. Quasi-graphic matroids
  5. Intermediate matroids
  6. Bracelet functions and tripartitions of cycles
  7. Bracelet functions
  8. Proper tripartitions
  9. Degenerate tripartitions
  10. Examples
  11. On quasi-graphic matroids
  12. Representability over a field
  13. Minors
  14. Connectivity
  15. Bases, independent sets, cocircuits
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $M$ be a connected matroid and let $(G,\mathcal{B})$ be a biased graph with $E(G) = E(M)$. The following are equivalent.

Figures (5)

  • Figure 1: Every bridgeless graph of cyclomatic number three is a subdivision of a contraction of one of the graphs in this figure.
  • Figure 2: In a $2m\times 2m$ torus grid there is a bracelet in each of homology classes $(1,0),(1,1),\ldots,(1,m-1)$. In each torus grid above, the pair of cycles bounding the union of the shaded faces form a bracelet.
  • Figure 3: Example \ref{['ex:differingreps']}(a). Two graphs for a frame matroid.
  • Figure 4: Example \ref{['ex:differingreps']}(b).
  • Figure 5: Depictions of cocircuits from Parts (2), (3), and (4) of Theorem \ref{['thm:cocircuits']}.

Theorems & Definitions (94)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Example 1
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • ...and 84 more