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Cops and Robbers: A $\times$-homotopy Invariant Variant

Tien Chih, Laura Scull

Abstract

Cops and Robbers is a pursuit-evasion game played on graphs, of which many variants have been developed and studied. We introduce a variant of this game, "Sneaky-Active Cops and Robbers", where all cops and robber must move on their turn, and where the robber is allowed to move onto a cop position without being captured. We show that for reflexive graphs, this game is equivalent to the classical cops and robbers and that the cop number for a graph is invariant under $\times$-homotopy equivalence. We then develop further properties of this game, computing cop numbers for a number of graph families and developing results about the behavior of categorical and box products of graphs.

Cops and Robbers: A $\times$-homotopy Invariant Variant

Abstract

Cops and Robbers is a pursuit-evasion game played on graphs, of which many variants have been developed and studied. We introduce a variant of this game, "Sneaky-Active Cops and Robbers", where all cops and robber must move on their turn, and where the robber is allowed to move onto a cop position without being captured. We show that for reflexive graphs, this game is equivalent to the classical cops and robbers and that the cop number for a graph is invariant under -homotopy equivalence. We then develop further properties of this game, computing cop numbers for a number of graph families and developing results about the behavior of categorical and box products of graphs.
Paper Structure (9 sections, 28 theorems, 5 equations, 1 figure)

This paper contains 9 sections, 28 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Background on $\times$-Homomotopy
  4. Background on Cops and Robbers
  5. The basics of Sneaky-Active Cops and Robbers
  6. Cop numbers for particular graph families
  7. The Categorical Product
  8. The Box Product
  9. Future Directions

Key Result

Theorem 3.4

If $X, Y$ are $\times$-homotopy equivalent,then $c_{SA}(X) = c_{SA}(Y)$.

Figures (1)

  • Figure 1: A table of cop numbers.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.8
  • Definition 2.9
  • Definition 3.1
  • Example 3.2
  • ...and 64 more