{"paper":{"title":"Excluded vertex-minors for graphs of linear rank-width at most k","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jisu Jeong, O-joung Kwon, Sang-il Oum","submitted_at":"2013-11-11T21:38:15Z","abstract_excerpt":"Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each $k$, there is a finite obstruction set $\\mathcal{O}_k$ of graphs such that a graph $G$ has linear rank-width at most $k$ if and only if no vertex-minor of $G$ is isomorphic to a graph in $\\mathcal{O}_k$. However, no attempts have been made to bound the number of graphs in $\\mathcal{O}_k$ for $k\\ge 2$. We show that for each $k$, there are at least $2^{\\Omega(3^k)}$ pairwise locally non-equivalent graphs in $\\mathcal{O}_k$, and ther"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2618","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"}