{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/transducing-paths-in-graph-classes-with","title":"Transducing paths in graph classes with unbounded shrubdepth","arxiv_id":"2203.16900","date":"2022-03-31","proceeding":null,"authors":["Michał Pilipczuk","Patrice Ossona de Mendez","Sebastian Siebertz"],"abstract":"Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class $\\mathscr{C}$ can be $\\mathsf{FO}$-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from $\\mathscr{C}$ one cannot $\\mathsf{FO}$-transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the $\\mathsf{MSO}$-transduction quasi-order, even in the stronger form that concerns $\\mathsf{FO}$-transductions instead of $\\mathsf{MSO}$-transductions. The backbone of our proof is a graph-theoretic statement that says the following: If a graph $G$ excludes a path, the bipartite complement of a path, and a half-graph as semi-induced subgraphs, then the vertex set of $G$ can be partitioned into a bounded number of parts so that every part induces a cograph of bounded height, and every pair of parts semi-induce a bi-cograph of bounded height. This statement may be of independent interest; for instance, it implies that the graphs in question form a class that is linearly $\\chi$-bounded.","url_abs":"https://arxiv.org/abs/2203.16900v1","url_pdf":"https://arxiv.org/pdf/2203.16900v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"transducing-paths-in-graph-classes-with","repo_url":"https://github.com/sail-sg/inceptionnext","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}