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Blind cop-width and balanced minors of graphs

Hector Buffière, Rutger Campbell, Kevin Hendrey, Sang-il Oum

TL;DR

This paper studies the radius-1 blind cop-width bcw_1 as a central graph-structure parameter tied to pursuit-evasion games, showing its equivalence with the inspection number and linking it to treewidth and pathwidth through balanced minors. It introduces balanced minors and uses them to derive strong lower bounds on bcw_1 via cliques, grids, and complete binary trees, ultimately proving that classes with bounded radius-1 bcw have bounded treewidth. A key triumph is establishing that the topological blind cop-width is Theta(treewidth) and that every graph of treewidth k has a subdivision with bcw_1 ≤ k+3, using a detailed subdivision-length construction. The paper also investigates blind flip-width, disproves a conjecture about dense-graph characterizations, and situates bcw_1 within a web of related parameters such as the inspection number, zero-visibility cop number, and hunting number, highlighting both their connections and limits. Collectively, these results illuminate the structural underpinnings of blind pursuit in graphs and hint at future directions for precise characterizations and algorithmic applications.

Abstract

We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in the graph. The blind cop-width is the minimum number of cops needed to catch the robber on a given graph. We link it with other known graph parameters defined in terms of pursuit-evasion games, and show a new lower bound with respect to treewidth. The proof introduces the notion of balanced minors, where all branch sets of a minor model have equal size.

Blind cop-width and balanced minors of graphs

TL;DR

This paper studies the radius-1 blind cop-width bcw_1 as a central graph-structure parameter tied to pursuit-evasion games, showing its equivalence with the inspection number and linking it to treewidth and pathwidth through balanced minors. It introduces balanced minors and uses them to derive strong lower bounds on bcw_1 via cliques, grids, and complete binary trees, ultimately proving that classes with bounded radius-1 bcw have bounded treewidth. A key triumph is establishing that the topological blind cop-width is Theta(treewidth) and that every graph of treewidth k has a subdivision with bcw_1 ≤ k+3, using a detailed subdivision-length construction. The paper also investigates blind flip-width, disproves a conjecture about dense-graph characterizations, and situates bcw_1 within a web of related parameters such as the inspection number, zero-visibility cop number, and hunting number, highlighting both their connections and limits. Collectively, these results illuminate the structural underpinnings of blind pursuit in graphs and hint at future directions for precise characterizations and algorithmic applications.

Abstract

We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in the graph. The blind cop-width is the minimum number of cops needed to catch the robber on a given graph. We link it with other known graph parameters defined in terms of pursuit-evasion games, and show a new lower bound with respect to treewidth. The proof introduces the notion of balanced minors, where all branch sets of a minor model have equal size.

Paper Structure

This paper contains 25 sections, 39 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minors and subdivisions.
  4. Sparse graph parameters
  5. Treewidth.
  6. Pathwidth.
  7. Properties.
  8. Cops and robber games
  9. Blind variant.
  10. Finite radius does not matter much for blind cops
  11. Balanced minors and lower bounds
  12. Lower bound lemma
  13. Complete graph as a balanced minor
  14. Highly connected minors with many cliques
  15. Outerplanar graph as a balanced minor
  16. ...and 10 more sections

Key Result

Proposition 2.1

The complete binary tree of height $n$ has pathwidth $\lceil n/2\rceil$.

Figures (3)

  • Figure 1: A graph having radius-$1$ blind cop-width $6$ or $7$ and containing $K_8$ as a balanced minor. Thick red edges have been subdivided $2^{12}$ times.
  • Figure 2: A binary tree as a balanced minor of a grid minor model.
  • Figure 3: The state of the game when the cops are cleaning the first subtree.

Theorems & Definitions (69)

  • Conjecture 1.1: note=Toruńczyk T23,store=torunczyk
  • Proposition 2.1: Scheffler Scheffler1989
  • Definition 2.2: Helicopter games
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 59 more