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Cops and Robber -- When Capturing is not Surrounding

Paul Jungeblut, Samuel Schneider, Torsten Ueckerdt

Abstract

We consider "surrounding" versions of the classic Cops and Robber game. The game is played on a connected graph in which two players, one controlling a number of cops and the other controlling a robber, take alternating turns. In a turn, each player may move each of their pieces: The robber always moves between adjacent vertices. Regarding the moves of the cops we distinguish four versions that differ in whether the cops are on the vertices or the edges of the graph and whether the robber may move on/through them. The goal of the cops is to surround the robber, i.e., occupying all neighbors (vertex version) or incident edges (edge version) of the robber's current vertex. In contrast, the robber tries to avoid being surrounded indefinitely. Given a graph, the so-called cop number denotes the minimum number of cops required to eventually surround the robber. We relate the different cop numbers of these versions and prove that none of them is bounded by a function of the classical cop number and the maximum degree of the graph, thereby refuting a conjecture by Crytser, Komarov and Mackey [Graphs and Combinatorics, 2020].

Cops and Robber -- When Capturing is not Surrounding

Abstract

We consider "surrounding" versions of the classic Cops and Robber game. The game is played on a connected graph in which two players, one controlling a number of cops and the other controlling a robber, take alternating turns. In a turn, each player may move each of their pieces: The robber always moves between adjacent vertices. Regarding the moves of the cops we distinguish four versions that differ in whether the cops are on the vertices or the edges of the graph and whether the robber may move on/through them. The goal of the cops is to surround the robber, i.e., occupying all neighbors (vertex version) or incident edges (edge version) of the robber's current vertex. In contrast, the robber tries to avoid being surrounded indefinitely. Given a graph, the so-called cop number denotes the minimum number of cops required to eventually surround the robber. We relate the different cop numbers of these versions and prove that none of them is bounded by a function of the classical cop number and the maximum degree of the graph, thereby refuting a conjecture by Crytser, Komarov and Mackey [Graphs and Combinatorics, 2020].
Paper Structure (16 sections, 13 theorems, 11 equations, 4 figures)

This paper contains 16 sections, 13 theorems, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Trivial Bounds.
  4. Related Work
  5. Outline of the Paper
  6. Relating the Different Versions
  7. Explicit Graphs and Constructions
  8. Complete Bipartite Graphs
  9. Regular Graphs with Leaves
  10. Graphs from Mutually Orthogonal Latin Squares
  11. Line Graphs of Complete Graphs
  12. When Capturing is not Surrounding
  13. The Graph H[s].
  14. The Graph H[s,l].
  15. The Graph H[s,l,m].
  16. ...and 1 more sections

Key Result

theorem 1

There is an infinite family of connected graphs $G$ with classical cop number $c(G) = 2$ and $\Delta(G) = 3$ while neither $c_V(G)$, $c_{V, \mathrm{r}}(G)$, $c_E(G)$ nor $c_{E, \mathrm{r}}(G)$ can be bounded by any function of $c(G)$ and the maximum degree $\Delta(G)$.

Figures (4)

  • Figure 1: Left: Two Latin squares and their juxtaposition, proving that they are orthogonal. Right: The graph $G_k$ created from $k-1$ MOLS of order $k$. The vertices in $R$ correspond to the rows of $A$, the middle vertices correspond to the cells of $A$ (ordered row by row in the drawing) and the vertices in $\mathcal{L}$ correspond to the parts of the MOLS.
  • Figure 2: Iterating $C_6 = H_2$ to obtain $r$-regular (bipartite) graphs $H_r$ with $\mathop{\mathrm{girth}}\nolimits(H_r) \geq 5$.
  • Figure 3: Construction of $H[s,\ell]$ based on $H[s]$. A directed edge $ab$ in $H[s]$ and the corresponding trees $T^\mathrm{out}(a)$, $T^\mathrm{in}(b)$, and path $P(ab)$ with middle edge $e(ab)$ in $H[s,\ell]$.
  • Figure 4: Construction of $H[s,\ell,m]$ based on two copies of $H[s,\ell]$. A directed edge $ab$ in $H[s]$ and the corresponding sets $S(a)$, $S(b)$, $F_2(a)$, etc. and edge $e(ab)$ in $H[s,\ell,m]$.

Theorems & Definitions (25)

  • theorem 1
  • theorem 2
  • proof : Proof of \ref{['thm:bounds']} (Upper Bounds)
  • corollary 1
  • proof
  • proposition 1
  • proof
  • lemma 1
  • proof
  • corollary 2
  • ...and 15 more