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Cops against a cheating robber

Nancy E. Clarke, Danny Dyer, William Kellough

TL;DR

This paper studies the cheating-robot variant of Cops and Robber by defining the cheating robot number $c_{cr}(G)$ and introducing the push number $p_{cr}(G)$ to quantify cops' pushing actions. It builds a framework connecting $c_{cr}(G)$ to the surrounding number $\sigma(G)$ via the inequality $c_{cr}(G) \le \sigma(G) \le c_{cr}(G) + p_{cr}(G)$, and develops a polynomial-time method to decide fixed-$k$ instances of $c_{cr}(G)$, enabling tractable analysis. The work applies these concepts to planar graphs (with tight results for bipartite planar graphs) and to graph products (providing bounds and exact values for several families), thereby enriching understanding of pursuit-evasion dynamics under reactive opponents. It also sketches several open problems, such as the universal bound on $p_{cr}(G)$ and maximum possible $c_{cr}(G)$ for planar graphs, guiding directions for further research.

Abstract

We investigate a cheating robot version of Cops and Robber, first introduced by Huggan and Nowakowski, where both the cops and the robber move simultaneously, but the robber is allowed to react to the cops' moves. For conciseness, we refer to this game as Cops and Cheating Robot. The cheating robot number for a graph is the fewest number of cops needed to win on the graph. We introduce a new parameter for this variation, called the push number, which gives the value for the minimum number of cops that move onto the robber's vertex given that there are a cheating robot number of cops on the graph. After producing some elementary results on the push number, we use it to give a relationship between Cops and Cheating Robot and Surrounding Cops and Robbers. We investigate the cheating robot number for planar graphs and give a tight bound for bipartite planar graphs. We show that determining whether a graph has a cheating robot number at most fixed $k$ can be done in polynomial time. We also obtain bounds on the cheating robot number for strong and lexicographic products of graphs.

Cops against a cheating robber

TL;DR

This paper studies the cheating-robot variant of Cops and Robber by defining the cheating robot number and introducing the push number to quantify cops' pushing actions. It builds a framework connecting to the surrounding number via the inequality , and develops a polynomial-time method to decide fixed- instances of , enabling tractable analysis. The work applies these concepts to planar graphs (with tight results for bipartite planar graphs) and to graph products (providing bounds and exact values for several families), thereby enriching understanding of pursuit-evasion dynamics under reactive opponents. It also sketches several open problems, such as the universal bound on and maximum possible for planar graphs, guiding directions for further research.

Abstract

We investigate a cheating robot version of Cops and Robber, first introduced by Huggan and Nowakowski, where both the cops and the robber move simultaneously, but the robber is allowed to react to the cops' moves. For conciseness, we refer to this game as Cops and Cheating Robot. The cheating robot number for a graph is the fewest number of cops needed to win on the graph. We introduce a new parameter for this variation, called the push number, which gives the value for the minimum number of cops that move onto the robber's vertex given that there are a cheating robot number of cops on the graph. After producing some elementary results on the push number, we use it to give a relationship between Cops and Cheating Robot and Surrounding Cops and Robbers. We investigate the cheating robot number for planar graphs and give a tight bound for bipartite planar graphs. We show that determining whether a graph has a cheating robot number at most fixed can be done in polynomial time. We also obtain bounds on the cheating robot number for strong and lexicographic products of graphs.
Paper Structure (7 sections, 27 theorems, 13 equations, 4 figures, 1 algorithm)

This paper contains 7 sections, 27 theorems, 13 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Technical Introduction
  3. The Push Number
  4. Planar Graphs
  5. Complexity
  6. Graph Products
  7. Further Directions

Key Result

Theorem 2.1

HN21 If $G$ is a graph with a $k$-core where $k\in \mathbb{Z}^+$, then $c_{cr}(G)\geq k$.

Figures (4)

  • Figure 1: A graph $G$ with a cheating robot number of two that contains a subgraph $H$ with fewer edges and a cheating robot number of at least three.
  • Figure 2: A graph illustrating that the converse of Theorem \ref{['thm: many vertices of high degree implies nonzero push number']} does not hold.
  • Figure 3: A graph $G$ such that $c_{cr}(G) = \sigma(G)$ but $p_{cr}(G) > 0$.
  • Figure 4: The graph $\mathop{\mathrm{\text{DS}}}\nolimits(I_{20})$.

Theorems & Definitions (45)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • ...and 35 more