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Unavoidable cycle-contraction minors of large $2$-connected graphs

Wayne Ge, James Oxley

TL;DR

This work develops a dual to the classical unavoidable induced-subgraph results by identifying unavoidable cycle-contraction minors (cc-minors) in sufficiently large loopless 2-connected graphs. It introduces the notion of $r$-templates and a graph-labeled tree framework, and proves that large graphs necessarily contain cc-minors that are parallel-path extensions of templates with many parts. The analysis moves from 2-edge-connected to 3-connected graphs, leveraging Tutte-style decompositions and Ramsey-type results for large 3-connected graphs to establish the unavoidable cc-minor structure. The matroid perspective extends the results to induced restrictions in regular matroids, highlighting the duality with graphic and cographic matroids and underscoring the broader relevance to matroid theory.

Abstract

It is well known that every sufficiently large connected graph has, as an induced subgraph, $K_n$, $K_{1,n}$, or an $n$-vertex path. A 2023 paper of Allred, Ding, and Oporowski identified a set of unavoidable induced subgraphs of sufficiently large $2$-connected graphs. In this paper, we establish a dual version of this theorem by focusing on the minors obtained by contracting cycles.

Unavoidable cycle-contraction minors of large $2$-connected graphs

TL;DR

This work develops a dual to the classical unavoidable induced-subgraph results by identifying unavoidable cycle-contraction minors (cc-minors) in sufficiently large loopless 2-connected graphs. It introduces the notion of -templates and a graph-labeled tree framework, and proves that large graphs necessarily contain cc-minors that are parallel-path extensions of templates with many parts. The analysis moves from 2-edge-connected to 3-connected graphs, leveraging Tutte-style decompositions and Ramsey-type results for large 3-connected graphs to establish the unavoidable cc-minor structure. The matroid perspective extends the results to induced restrictions in regular matroids, highlighting the duality with graphic and cographic matroids and underscoring the broader relevance to matroid theory.

Abstract

It is well known that every sufficiently large connected graph has, as an induced subgraph, , , or an -vertex path. A 2023 paper of Allred, Ding, and Oporowski identified a set of unavoidable induced subgraphs of sufficiently large -connected graphs. In this paper, we establish a dual version of this theorem by focusing on the minors obtained by contracting cycles.

Paper Structure

This paper contains 13 sections, 23 theorems, 1 equation, 16 figures.

Table of Contents

  1. Introduction
  2. Unavoidable families
  3. cc-minors
  4. A matroid perspective
  5. Preliminaries
  6. $r$-templates
  7. cc-minors of graphs
  8. Internally disjoint $XY$-paths
  9. cc-minors of $2$-edge-connected graphs
  10. Classes closed under cc-minors
  11. cc-minors of $3$-connected graphs
  12. cc-minors of large $3$-connected graphs
  13. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $r$ be a positive integer. There is an integer $R(r)$ such that every graph on at least $R(r)$ vertices has, as an induced subgraph, either $K_r$ or its complement $\overline{K_r}$.

Figures (16)

  • Figure 1: A $2$-connected graph $G$ and its tree decomposition
  • Figure 2: Three examples of fan-type graphs
  • Figure 3: Possible parts of templates
  • Figure 4: A sample $6$-template
  • Figure 5: Possible simplifications of $H$ that fall under cases (ii) and (iii).
  • ...and 11 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • proof
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • ...and 24 more