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Binary matroids and degree-boundedness for pivot-minors

Rutger Campbell, James Davies, Robert Hickingbotham

Abstract

We prove that for every bipartite graph $H$ and positive integer $s$, the class of $K_{s,s}$-subgraph-free graphs excluding $H$ as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every $K_{s,t}$-free bipartite circle graph with $s\le t$ has a vertex of degree at most $\max\{2s-2, t-1\}$ and provide examples showing that this is tight.

Binary matroids and degree-boundedness for pivot-minors

Abstract

We prove that for every bipartite graph and positive integer , the class of -subgraph-free graphs excluding as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every -free bipartite circle graph with has a vertex of degree at most and provide examples showing that this is tight.

Paper Structure

This paper contains 5 sections, 16 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fundamental graphs
  3. Induced subgraphs with large average degree
  4. Pivot-minors and binary matroids
  5. Open problems

Key Result

Theorem 1

For every bipartite graph $H$, there exists a function $f_H$ such that for every positive integer $s$, every $K_{s,s}$-free graph excluding $H$ as pivot-minor has average degree at most $f_H(s)$.

Figures (2)

  • Figure 1: Representations of bipartite circle graphs as fundamental graphs of planar graphs: black edges correspond to the spanning tree $T$, red edges are the non-tree edges $E(G)-E(T)$.
  • Figure 2: Pivoting the edge $xy$ in $G$ corresponds to edge-complementation within the three dashed regions and swapping the labels of $x$ and $y$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Theorem 4: de Fraysseix de1981local
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 15 more