{"paper":{"title":"Polynomial Gap Extensions of the Erd\\H{o}s-P\\'osa Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Dimitrios M. Thilikos, Jean-Florent Raymond","submitted_at":"2013-05-31T12:29:13Z","abstract_excerpt":"Given a graph $H$, we denote by ${\\cal M}(H)$ all graphs that can be contracted to $H$. The following extension of the Erd\\H{o}s-P\\'osa Theorem holds: for every $h$-vertex planar graph $H$, there exists a function $f_{H}$ such that every graph $G$, either contains $k$ disjoint copies of graphs in ${\\cal M}(H)$, or contains a set of $f_{H}(k)$ vertices meeting every subgraph of $G$ that belongs in ${\\cal M}(H)$. In this paper we prove that this is the case for every graph $H$ of pathwidth at most 2 and, in particular, that $f_{H}(k) = 2^{O(h^2)}\\cdot k^{2}\\cdot \\log k$. As a main ingredient of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.7376","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":""},"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"}