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$k$-Hyperopic Cops and Robber

Nicholas Crawford, Vesna Iršič Chenoweth

TL;DR

A generalization of hyperopic cops and robber, analogous to the $k$-visibility cops and robber, is introduced and an upper bound in terms of the number of vertices of the graph is presented for $k \geq 3$.

Abstract

A generalization of hyperopic cops and robber, analogous to the $k$-visibility cops and robber, is introduced in this paper. For a positive integer $k$ the $k$-hyperopic game of cops and robber is defined similarly as the usual cops and robber game, but with the robber being omniscient and invisible to the cops that are at distance at most $k$ away from the robber. The cops win the game if, after a finite number of rounds, a cop occupies the same vertex as robber. Otherwise, robber wins. The minimum number of cops needed to win the game on a graph $G$ is the $k$-hyperopic cop number $c_{H,k}(G)$ of $G$. In addition to basic properties of the new invariant, cop-win graphs are characterized and a general upper bound in terms of the matching number of the graph is given. The invariant is also studied on trees where the upper bounds mostly depend on the relation between $k$ and the diameter of the tree. It is also proven that the 2-hyperopic cop number of outerplanar graphs is at most 2 and an upper bound in terms of the number of vertices of the graph is presented for $k \geq 3$.

$k$-Hyperopic Cops and Robber

TL;DR

A generalization of hyperopic cops and robber, analogous to the -visibility cops and robber, is introduced and an upper bound in terms of the number of vertices of the graph is presented for .

Abstract

A generalization of hyperopic cops and robber, analogous to the -visibility cops and robber, is introduced in this paper. For a positive integer the -hyperopic game of cops and robber is defined similarly as the usual cops and robber game, but with the robber being omniscient and invisible to the cops that are at distance at most away from the robber. The cops win the game if, after a finite number of rounds, a cop occupies the same vertex as robber. Otherwise, robber wins. The minimum number of cops needed to win the game on a graph is the -hyperopic cop number of . In addition to basic properties of the new invariant, cop-win graphs are characterized and a general upper bound in terms of the matching number of the graph is given. The invariant is also studied on trees where the upper bounds mostly depend on the relation between and the diameter of the tree. It is also proven that the 2-hyperopic cop number of outerplanar graphs is at most 2 and an upper bound in terms of the number of vertices of the graph is presented for .

Paper Structure

This paper contains 6 sections, 18 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic properties
  3. General upper bound
  4. Trees
  5. Outerplanar
  6. Future Work

Key Result

Corollary 2.3

If $k \geq 2$ and $n \geq m \geq 1$, then $c_{H,k}(K_{m,n}) = m$.

Figures (4)

  • Figure 1: The tree $\widehat{T}.$
  • Figure 2: A schematic drawing of the graph $G_k$.
  • Figure 3: A construction of a tree in $\mathcal{T}_{m+1}$ from $T_1, T_2, T_3 \in \mathcal{T}_m$.
  • Figure 4: A tree with diameter 10 and 9-hyperopic cop number equal to 3.

Theorems & Definitions (36)

  • proof
  • proof
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Example 2.6
  • Theorem 2.7
  • proof
  • ...and 26 more