{"paper":{"title":"Excluding a rectangular grid","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Cl\\'ement Rambaud","submitted_at":"2025-01-20T17:29:53Z","abstract_excerpt":"For every positive integer $k$, we define the $k$-treedepth as the largest graph parameter $\\mathrm{td}_k$ satisfying (i) $\\mathrm{td}_k(\\emptyset)=0$; (ii) $\\mathrm{td}_k(G) \\leq 1+ \\mathrm{td}_k(G-u)$ for every graph $G$ and every vertex $u \\in V(G)$; and (iii) if $G$ is a $(<k)$-clique-sum of $G_1$ and $G_2$, then $\\mathrm{td}_k(G) \\leq \\max \\{\\mathrm{td}_k(G_1),\\mathrm{td}_k(G_2)\\}$, for all graphs $G_1,G_2$. This parameter coincides with treedepth if $k=1$, and with treewidth plus $1$ if $k \\geq |V(G)|$. We prove that for every positive integer $k$, a class of graphs $\\mathcal{C}$ has bou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.11617","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.11617/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}