{"paper":{"title":"Obstructions to Erd\\H{o}s-P\\'osa Dualities for Minors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.CO","authors_text":"Christophe Paul, Dimitrios M. Thilikos, Evangelos Protopapas, Sebastian Wiederrecht","submitted_at":"2024-07-12T20:08:19Z","abstract_excerpt":"Let ${\\cal G}$ and ${\\cal H}$ be minor-closed graph classes. The pair $({\\cal H},{\\cal G})$ is an Erd\\H{o}s-P\\'osa pair (EP-pair) if there is a function $f$ where, for every $k$ and every $G\\in{\\cal G},$ either $G$ has $k$ pairwise vertex-disjoint subgraphs not belonging to ${\\cal H},$ or there is a set $S\\subseteq V(G)$ where $|S|\\leq f(k)$ and $G-S\\in{\\cal H}.$ The classic result of Erd\\H{o}s and P\\'osa says that if $\\mathcal{F}$ is the class of forests, then $({\\cal F},{\\cal G})$ is an EP-pair for every ${\\cal G}$. The class ${\\cal G}$ is an EP-counterexample for ${\\cal H}$ if ${\\cal G}$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.09671","kind":"arxiv","version":2},"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/2407.09671/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"}