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A note on highly connected $K_{2,\ell}$-minor free graphs

Nicolas Bousquet, Théo Pierron, Alexandra Wesolek

TL;DR

In $3$-connected $K_{2,\ell}$-minor-free graphs, the paper derives a sharp maximum-degree bound and a twin-free bounded-size result under high minimum degree without invoking Ding's decomposition, using a self-contained framework based on Steiner trees and 2-nested cuts. The main technical contributions show that any graph with $\delta\ge4$ has $\Delta\le7\ell$, and that $\delta\ge5$ with no twins of degree $5$ yields bounded size, broadening the landscape of structure theorems for minor-free graphs. The methods provide a direct way to deduce density and size constraints and connect to known edge-density bounds, offering insights for algorithmic applications on such graph classes.

Abstract

We show that every $3$-connected $K_{2,\ell}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\ell$. As a consequence, we show that every 3-connected $K_{2,\ell}$-minor free graph with minimum degree at least $5$ and no twins of degree $5$ has bounded size. Our proofs use Steiner trees and nested cuts; in particular, they do not rely on Ding's characterization of $K_{2,\ell}$-minor free graphs.

A note on highly connected $K_{2,\ell}$-minor free graphs

TL;DR

In -connected -minor-free graphs, the paper derives a sharp maximum-degree bound and a twin-free bounded-size result under high minimum degree without invoking Ding's decomposition, using a self-contained framework based on Steiner trees and 2-nested cuts. The main technical contributions show that any graph with has , and that with no twins of degree yields bounded size, broadening the landscape of structure theorems for minor-free graphs. The methods provide a direct way to deduce density and size constraints and connect to known edge-density bounds, offering insights for algorithmic applications on such graph classes.

Abstract

We show that every -connected -minor free graph with minimum degree at least has maximum degree at most . As a consequence, we show that every 3-connected -minor free graph with minimum degree at least and no twins of degree has bounded size. Our proofs use Steiner trees and nested cuts; in particular, they do not rely on Ding's characterization of -minor free graphs.
Paper Structure (3 sections, 3 theorems, 2 figures)

This paper contains 3 sections, 3 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Maximum degree
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

All $3$-connected $K_{2,\ell}$-minor free graphs with minimum degree $5$ containing no twins of degree $5$ have bounded size.

Figures (2)

  • Figure 1: An infinite family of twin-free $4$-connected and $4$-regular graphs without $K_{2,5}$-minor.
  • Figure 2: Contracting the red thick edges reduces a $2\times 4$ king's graph to a $2\times 3$ king's graph.

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Claim 2.2
  • proof
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}