Monotonicity of the cops and robber game for bounded depth treewidth
Isolde Adler, Eva Fluck
TL;DR
The paper investigates a variant of the cops and robber game with a bound $q$ on the number of cop placements, proving that a winning strategy with $k$ cops can be made monotone. This yields a new characterisation of graphs with bounded depth treewidth and shows these classes are homomorphism distinguishing closed, answering an open question. The authors introduce a matroid-inspired pre-decomposition and a breadth-first cleaning procedure to transform strategies while tightly controlling width and depth, providing a novel approach to monotonicity that extends to classic treewidth and treedepth settings. The results connect game-theoretic characterisations with homomorphism counts, offering techniques that may be useful for future work on width parameters and logical definability in graph classes.
Abstract
We study a variation of the cops and robber game characterising treewidth, where in each play at most q cops can be placed in order to catch the robber, where q is a parameter of the game. We prove that if k cops have a winning strategy in this game, then k cops have a monotone winning strategy. As a corollary we obtain a new characterisation of bounded depth treewidth, and we give a positive answer to an open question by Fluck, Seppelt and Spitzer (2024), thus showing that graph classes of bounded depth treewidth are homomorphism distinguishing closed. Our proof of monotonicity substantially reorganises a winning strategy by first transforming it into a pre-decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first "cleaning up" procedure along the pre-decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of cop placements simultaneously across all branches of the decomposition via a vertex exchange argument. We believe this can be useful in future research.