2026-07-25T01:46:56Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20562 2024-08-23T05:50:28Z ddc:004 open_access

Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth Adler, Isolde Fluck, Eva tree decompositions treewidth treedepth cops-and-robber game monotonicity homomorphism distinguishing closure We study a variation of the cops and robber game characterising treewidth, where in each round at most one cop may be placed and in each play at most q rounds are played, 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-tree decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first "cleaning up" procedure along the pre-tree decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of rounds simultaneously across all branches of the decomposition via a vertex exchange argument. We believe this can be useful in future research. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Isolde Adler and Eva Fluck 2024 Is Part Of LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2024.6 urn:nbn:de:0030-drops-205621 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.6 eng https://creativecommons.org/licenses/by/4.0/legalcode