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Cops and Attacking Robbers with Cycle Constraints

Alexander Clow, Melissa A. Huggan, M. E. Messinger

TL;DR

This work advances the study of the Cops and Attacking Robbers variant by delivering a structural characterization of triangle-free graphs with attacking cop number at most 2, leveraging domination-elimination concepts. It proves a tight bound for bipartite planar graphs, showing $cc(G)\le 4$ with a constructive tight example, and develops guardability lemmas for geodesic paths to support planar strategies. The authors also demonstrate a concrete separation between the attacking cop number and the classical cop number by constructing 17 graphs of order 58 with $cc(H)=6$ and $c(H)=3$, establishing $cc(H)-c(H)\ge 3$, and provide a lower-bound framework via $G^2-E$ that informs these separations. The paper closes with conjectures and open problems that chart a rich research agenda on triangle-free, planar, and general graphs in pursuit–evasion settings. The results offer new tools and directions for understanding how attack power interacts with traditional guarding and domination parameters in graph pursuit games.

Abstract

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph $G$, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by $\attCop(G)$. We characterise the triangle-free graphs $G$ with $\attCop(G) \leq 2$ via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs $G$ have $\attCop(G) \leq 4$ and show this is tight by constructing a bipartite planar graph $G$ with $\attCop(G) = 4$. Finally we construct $17$ non-isomorphic graphs $H$ of order $58$ with $\attCop(H) = 6$ and $\cop(H)=3$. This provides the first example of a graph $H$ with $\attCop(H) - \cop(H) \geq 3$ extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Prałat, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems.

Cops and Attacking Robbers with Cycle Constraints

TL;DR

This work advances the study of the Cops and Attacking Robbers variant by delivering a structural characterization of triangle-free graphs with attacking cop number at most 2, leveraging domination-elimination concepts. It proves a tight bound for bipartite planar graphs, showing with a constructive tight example, and develops guardability lemmas for geodesic paths to support planar strategies. The authors also demonstrate a concrete separation between the attacking cop number and the classical cop number by constructing 17 graphs of order 58 with and , establishing , and provide a lower-bound framework via that informs these separations. The paper closes with conjectures and open problems that chart a rich research agenda on triangle-free, planar, and general graphs in pursuit–evasion settings. The results offer new tools and directions for understanding how attack power interacts with traditional guarding and domination parameters in graph pursuit games.

Abstract

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph , the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by . We characterise the triangle-free graphs with via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs have and show this is tight by constructing a bipartite planar graph with . Finally we construct non-isomorphic graphs of order with and . This provides the first example of a graph with extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Prałat, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems.
Paper Structure (5 sections, 10 theorems, 6 equations, 5 figures)

This paper contains 5 sections, 10 theorems, 6 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Triangle-Free Graphs with $\mathop{\mathrm{cc}}\nolimits(G) \leq 2$.
  3. Planar Graphs
  4. Constructing Graphs where $\mathop{\mathrm{cc}}\nolimits(H)-\mathop{\mathrm{c}}\nolimits(H)=3$
  5. Future Work

Key Result

Lemma 2.2

Let $G = (V,E)$ be a triangle-free graph with $\mathop{\mathrm{cc}}\nolimits(G) \geq 2$. If $u \in V$ is a dominated vertex in $G$ and $\gamma(G-u)\geq 2$, then $\mathop{\mathrm{cc}}\nolimits(G - u) = \mathop{\mathrm{cc}}\nolimits(G)$.

Figures (5)

  • Figure 1: The graph $P_4$ (left), $C_7$ (middle), and $G$ (right). We note that $\mathop{\mathrm{c}}\nolimits(P_4)=1 = \mathop{\mathrm{c}}\nolimits(G)$ and $\mathop{\mathrm{c}}\nolimits(C_7)=2$, while $\mathop{\mathrm{cc}}\nolimits(G) = 1$, $\mathop{\mathrm{cc}}\nolimits(P_4)=2$, and $\mathop{\mathrm{cc}}\nolimits(C_7)=3$.
  • Figure 2: An example of a graph $G$ with at least one triangle, no dominated vertices, and domination number three such that $\mathop{\mathrm{cc}}\nolimits(G)=2$.
  • Figure 3: The dodecahedral graph subdivided exactly once on every edge.
  • Figure 4: (top) is visualization of the start of Case (ii); (bottom) is the visualization of the later part of Case (ii).
  • Figure 5: The graph $G_1$ (left) from biggs1980trivalentbrinkmann1995smallest and the graph $H_1 = G_1^2 - E(G_1)$ (right).

Theorems & Definitions (23)

  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2: bonato2014robber Theorem 3
  • Lemma 3.3
  • ...and 13 more