{"paper":{"title":"The excluded minors for isometric realizability in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.MG","authors_text":"Antonios Varvitsiotis, Gwena\\\"el Joret, Samuel Fiorini, Tony Huynh","submitted_at":"2015-11-25T13:39:08Z","abstract_excerpt":"Let $G$ be a graph and $p \\in [1, \\infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \\in V(G)} \\subseteq \\mathbb{R}^m$, there exist vectors $(q_v)_{v \\in V(G)} \\subseteq \\mathbb{R}^k$ satisfying $$\\|r_v-r_w\\|_p=\\|q_v-q_w\\|_p, \\ \\text{ for all }\\ vw\\in E(G).$$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property $f_p(G) \\leq k$.\n  In this paper, we determine the complete set of excluded minors for $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08054","kind":"arxiv","version":5},"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"}