{"paper":{"title":"The graphs behind Reuleaux polyhedra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Edgardo Rold\\'an-Pensado, Eric Pauli, Luis Montejano, Miguel Raggi","submitted_at":"2019-04-29T15:06:13Z","abstract_excerpt":"This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, $3$-connected and strongly self-dual. We study the question of when these conditions are sufficient.\n  If $G$ is any such a graph with isomorphism $\\tau : G \\to G^*$ (where $G^*$ is the unique dual graph), a metric mapping is a map $\\eta : V(G) \\to \\mathbb R^3$ such that the diameter of $\\eta(G)$ is $1$ and for every pair of vertices $(u,v)$ such that $u\\in \\tau(v)$ we have dist$(\\eta(u),\\eta(v)) = 1$. If $\\eta$ is injective, it is called a metric embedding. Note that a metric embedding gives ris"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.12761","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"}