Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved bounds on the cop number when forbidding a minor

Franklin Kenter, Erin Meger, Jérémie Turcotte

TL;DR

This work advances bounds on the cop number for graphs excluding a fixed minor by extending Andreae’s path-guarding approach through a decomposition framework of the minor. The authors define a detailed minor-constructible state model with bags $A_w$ and guarded interconnecting paths $Q_P$, establishing the main inequality $c(G) \le \mathds{1}_{\ell}+\sum_{P\in \P} \lceil \ell_P/3 \rceil$ for any connected $H$-minor-free graph, where $\ell_P$ encodes the contribution of each path in the decomposition. They further present simplified corollaries and apply the theorem to concrete families—$K_{3,t}$- and $K_{2,t}$-minor-free graphs, complete graphs, and linklessly embeddable graphs—obtaining notably improved bounds (e.g., $c(G) \le 6$ for linklessly embeddable graphs and $c(G) \le t$ for $K_{3,t}$-minor-free graphs). The results illuminate how subdivision- or sparsity-driven minor structures yield substantial gains over prior bounds and offer a versatile toolkit for analyzing cop numbers in minor-closed graph classes. The paper also outlines avenues for further tightening bounds, exploring multiple-minor obstructions, and constructing new lower bounds to deepen understanding of cop numbers in various graph families.

Abstract

Andreae (1986) proved that the cop number of connected $H$-minor-free graphs is bounded for every graph $H$. In particular, the cop number is at most $|E(H-h)|$ if $H-h$ contains no isolated vertex, where $h\in V(H)$. The main result of this paper is an improvement on this bound, which is most significant when $H$ is small or sparse, for instance when $H-h$ can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of $K_{3,t}$-minor-free graphs, $K_{2,t}$-minor-free graphs and linklessly embeddable graphs.

Improved bounds on the cop number when forbidding a minor

TL;DR

This work advances bounds on the cop number for graphs excluding a fixed minor by extending Andreae’s path-guarding approach through a decomposition framework of the minor. The authors define a detailed minor-constructible state model with bags and guarded interconnecting paths , establishing the main inequality for any connected -minor-free graph, where encodes the contribution of each path in the decomposition. They further present simplified corollaries and apply the theorem to concrete families—- and -minor-free graphs, complete graphs, and linklessly embeddable graphs—obtaining notably improved bounds (e.g., for linklessly embeddable graphs and for -minor-free graphs). The results illuminate how subdivision- or sparsity-driven minor structures yield substantial gains over prior bounds and offer a versatile toolkit for analyzing cop numbers in minor-closed graph classes. The paper also outlines avenues for further tightening bounds, exploring multiple-minor obstructions, and constructing new lower bounds to deepen understanding of cop numbers in various graph families.

Abstract

Andreae (1986) proved that the cop number of connected -minor-free graphs is bounded for every graph . In particular, the cop number is at most if contains no isolated vertex, where . The main result of this paper is an improvement on this bound, which is most significant when is small or sparse, for instance when can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of -minor-free graphs, -minor-free graphs and linklessly embeddable graphs.
Paper Structure (13 sections, 14 theorems, 8 equations, 3 figures)

This paper contains 13 sections, 14 theorems, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Guarding paths
  4. Main result
  5. Key ideas of the proof
  6. Applications
  7. Simplified versions of the main result
  8. Recovering Andreae's results
  9. Complete bipartite graphs
  10. Complete graphs
  11. Linklessly embeddable graphs
  12. Greater improvement factor
  13. Future directions

Key Result

Theorem 1.1

andreae_pursuit_1986 Let $H$ be a graph and $h\in V(H)$ be a vertex such that $H-h$ has no isolated vertex. If $G$ is a connected $H$-minor-free graph, then $c(G) \leq |E(H-h)|$.

Figures (3)

  • Figure 1: Example of a decomposition of a graph $H$ and of a state of a game played on an $H$-minor-free graph $G$.
  • Figure 2: Petersen family.
  • Figure 3: The graphs $H_t$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • proof
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3
  • proof
  • Definition
  • ...and 39 more