{"paper":{"title":"The Facial Common Neighbourhood Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Riccardo W. Maffucci","submitted_at":"2026-05-15T17:35:45Z","abstract_excerpt":"Given a polyhedron (planar, $3$-connected graph) $G$, we investigate its common neighbourhood graph con($G$). For cubic ($3$-regular) polyhedra, we show that the planarity of con($G$) depends on the number of odd faces of $G$, and on their adjacency. We then prove that for all other polyhedra, con($G$) is non-planar.\n  We introduce a novel concept for polyhedra (and more generally, for plane graphs) $G$, namely the `facial common neighbourhood graph' facecon($G$). Its definition takes into account pairs of vertices with a common neighbour on the same face of $G$. It is a spanning subgraph of c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16224","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16224/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"shingle_duplication","ran_at":"2026-05-19T17:49:43.974414Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T17:49:43.634146Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:24.340123Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"external_links","ran_at":"2026-05-19T17:31:31.116127Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T17:22:05.400556Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.380095Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"621c5184c69f96c103f4cb0d271452292e60b868fd67bfb29614fc4b7c51b100"},"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"}