{"paper":{"title":"Linear-Vertex Kernel for the Problem of Packing $r$-Stars into a Graph without Long Induced Paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Anders Yeo, Bin Sheng, Florian Barbero, Gregory Gutin, Mark Jones","submitted_at":"2015-10-13T08:02:41Z","abstract_excerpt":"Let integers $r\\ge 2$ and $d\\ge 3$ be fixed. Let ${\\cal G}_d$ be the set of graphs with no induced path on $d$ vertices. We study the problem of packing $k$ vertex-disjoint copies of $K_{1,r}$ ($k\\ge 2$) into a graph $G$ from parameterized preprocessing, i.e., kernelization, point of view. We show that every graph $G\\in {\\cal G}_d$ can be reduced, in polynomial time, to a graph $G'\\in {\\cal G}_d$ with $O(k)$ vertices such that $G$ has at least $k$ vertex-disjoint copies of $K_{1,r}$ if and only if $G'$ has. Such a result is known for arbitrary graphs $G$ when $r=2$ and we conjecture that it ho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03564","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":""},"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"}