Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cops and robbers on $2K_2$-free graphs

Jérémie Turcotte

TL;DR

It is proved that the cop number of any $2K_2$-free graph is at most 2, which was previously conjectured by Sivaraman and Testa.

Abstract

We prove that the cop number of any $2K_2$-free graph is at most 2, proving a conjecture of Sivaraman and Testa. We also show that the upper bound of $3$ on the cop number of $2K_1+K_2$-free (co-diamond--free) graphs is best possible.

Cops and robbers on $2K_2$-free graphs

TL;DR

It is proved that the cop number of any -free graph is at most 2, which was previously conjectured by Sivaraman and Testa.

Abstract

We prove that the cop number of any -free graph is at most 2, proving a conjecture of Sivaraman and Testa. We also show that the upper bound of on the cop number of -free (co-diamond--free) graphs is best possible.

Paper Structure

This paper contains 6 sections, 19 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Traps
  3. Finding connected traps
  4. A winning strategy
  5. Co-diamond--free graphs
  6. Further directions

Key Result

Theorem 1.1

andreae_pursuit_1986 If $H$ is a graph, then there exists $M_H\in \mathbb{N}$ such that for any $H$-minor-free connected graph $G$ we have $c(G)\leq M_H$.

Figures (1)

  • Figure 1: $K_4\times K_4$: All pairs of vertices are adjacent except when aligned vertically or horizontally. Some edges overlap in the drawing.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • ...and 24 more