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Fast and Furious: A study on Monotonicity and Speed in Cops-and-Robber Games

Eva Fluck, David Philipps

TL;DR

The paper analyzes multiple Cops-and-Robber game variants parameterized by robber speed and by cop/robber monotonicity and visibility. It proves that several invisible variants precisely characterize path-width via copwidth measures, while visible variants exhibit unbounded monotonicity costs and separations from width notions like tree-width. It also connects robber-monotonicity with strong coloring numbers, establishing functional equivalences among lazy variants and bounding the robber-monotone width in terms of non-monotone width and speed; furthermore, it characterizes graph classes with bounded expansion through these width measures. Collectively, the results illuminate when monotonicity restrictions simplify width computations and how these game-theoretic notions reflect structural graph sparsity, with implications for algorithmic applications and complexity. The work also outlines open questions about specific speeds (e.g., $s=2,3$) and the precise behavior of the bounded-speed/monotonicity landscape.

Abstract

In this paper, we study different variants of the Cops-and-Robber game with respect to cop- and robber\-/monotonicity. We study a visible and invisible robber and variants where the robber is lazy, thus can only move when the cops announce to move on top of him. In all four combinations, we also vary the number $s$ of edges that the robber can traverse in a single round, called speed. We complete the study of the unbounded speed case by showing that, besides the active variants, also the visible lazy variant has both the cop- and robber\-/monotonicity property. Furthermore, we prove that the cop\-/monotone invisible lazy copwidth characterizes path-width, while the non\-/monotone and robber\-/monotone is known to characterize tree-width, thus these variants differ even in the unbounded speed case. We find that, even with speed restriction, the cop\-/monotone invisible copwidth and the robber\-/monotone invisible active copwidth all characterize path-width. On the other hand, we show that the path-width of a graph can be arbitrarily larger than the number of cops needed to win the non\-/monotone invisible active variant. To complete our study of cop\-/monotone variants, we show that also in the visible variants the cop\-/monotone copwidth can be arbitrarily larger than the non\-/monotone. Regarding robber\-/monotonicity, for all speeds $s\geq 4$, we give graphs where the non\-/monotone and robber\-/monotone copwidth differ. On the other hand, we prove that there is a function that bounds the robber\-/monotone copwidth in terms of the non\-/monotone copwidth and the speed, thus the gap between the variants is bounded. This proof also yields that a graph class has bounded expansion if and only if, for every speed $s$, the number of cops needed in any robber\-/monotone lazy variant is bounded by some constant $c(s)$.

Fast and Furious: A study on Monotonicity and Speed in Cops-and-Robber Games

TL;DR

The paper analyzes multiple Cops-and-Robber game variants parameterized by robber speed and by cop/robber monotonicity and visibility. It proves that several invisible variants precisely characterize path-width via copwidth measures, while visible variants exhibit unbounded monotonicity costs and separations from width notions like tree-width. It also connects robber-monotonicity with strong coloring numbers, establishing functional equivalences among lazy variants and bounding the robber-monotone width in terms of non-monotone width and speed; furthermore, it characterizes graph classes with bounded expansion through these width measures. Collectively, the results illuminate when monotonicity restrictions simplify width computations and how these game-theoretic notions reflect structural graph sparsity, with implications for algorithmic applications and complexity. The work also outlines open questions about specific speeds (e.g., ) and the precise behavior of the bounded-speed/monotonicity landscape.

Abstract

In this paper, we study different variants of the Cops-and-Robber game with respect to cop- and robber\-/monotonicity. We study a visible and invisible robber and variants where the robber is lazy, thus can only move when the cops announce to move on top of him. In all four combinations, we also vary the number of edges that the robber can traverse in a single round, called speed. We complete the study of the unbounded speed case by showing that, besides the active variants, also the visible lazy variant has both the cop- and robber\-/monotonicity property. Furthermore, we prove that the cop\-/monotone invisible lazy copwidth characterizes path-width, while the non\-/monotone and robber\-/monotone is known to characterize tree-width, thus these variants differ even in the unbounded speed case. We find that, even with speed restriction, the cop\-/monotone invisible copwidth and the robber\-/monotone invisible active copwidth all characterize path-width. On the other hand, we show that the path-width of a graph can be arbitrarily larger than the number of cops needed to win the non\-/monotone invisible active variant. To complete our study of cop\-/monotone variants, we show that also in the visible variants the cop\-/monotone copwidth can be arbitrarily larger than the non\-/monotone. Regarding robber\-/monotonicity, for all speeds , we give graphs where the non\-/monotone and robber\-/monotone copwidth differ. On the other hand, we prove that there is a function that bounds the robber\-/monotone copwidth in terms of the non\-/monotone copwidth and the speed, thus the gap between the variants is bounded. This proof also yields that a graph class has bounded expansion if and only if, for every speed , the number of cops needed in any robber\-/monotone lazy variant is bounded by some constant .

Paper Structure

This paper contains 15 sections, 20 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. The Game
  5. Path-Width and Invisible Variants
  6. Cop-Monotonicity and the Visible Variants
  7. Robber Monotonicity and Lazy Variants
  8. Bounded Expansion
  9. Conclusion
  10. Disclosure of Interests.
  11. Appendix
  12. $\mathrm{rm\hbox{[}1.0]{-} il\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)\geq\mathrm{rm\hbox{[}1.0]{-} vl\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)>10$
  13. $\mathrm{vl\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)\leq\mathrm{il\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)\leq10$
  14. Case $|\hat{Z}| \leq |Z|$:
  15. Case $|\hat{Z}| > |Z|$:

Key Result

theorem thmcountertheorem

For every graph $G$ and speed $s\in\mathbb{N}$, it holds that

Figures (6)

  • Figure 1: Known (black) and new (red, dashed) results for the variants. Arrows with open triangle heads represent a not strict inequality, arrows with filled triangle heads represent a strict inequality on some graphs. Lines represent equalities and arrows with a tip on the reverse side represent that the corresponding inequality is functionally bounded. $s$ is an arbitrary natural number greater than 1.
  • Figure 2: A similar graph to $G_4$ from \ref{['thmt@@TheoremVisActCopmonotonicity']}. In $G_4$ each node has 3 children, but the main ideas should become clear from this picture. Blue dashed lines represent paths of length $2s$. For \ref{['thmt@@TheoremVisLazyCopmonotonicity']} the length is $s+1$.
  • Figure 3: Relations between the colouring numbers and lazy copwidth variants for any $s>1$. Each arrow represents a strict inequality, note that the differences between the robber-/monotone and non-/monotone variants are only known for $s\geq 4$.
  • Figure 4: An example graph $G$ with $\mathrm{vl\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)\leq\mathrm{il\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)\leq10$ and $\mathrm{rm\hbox{[}1.0]{-} il\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)\geq\mathrm{rm\hbox{[}1.0]{-} vl\hbox{[}1.0]{-} \mathrm{cw_{4}}}(G)>10$. $K_n$ denotes the n-clique.
  • Figure 5: The red region shows the hiding spot of the robber. The blue circles are the cop-positions after the round. The blue verteces are places where the cops were placed in the last round and stay. Each dashed line represents a path of length $s$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • ...and 20 more