{"paper":{"title":"Nucleation-free $3D$ rigidity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"cs.CG","authors_text":"Ileana Streinu, Jialong Cheng, Meera Sitharam","submitted_at":"2013-11-19T20:06:38Z","abstract_excerpt":"When all non-edge distances of a graph realized in $\\mathbb{R}^{d}$ as a {\\em bar-and-joint framework} are generically {\\em implied} by the bar (edge) lengths, the graph is said to be {\\em rigid} in $\\mathbb{R}^{d}$. For $d=3$, characterizing rigid graphs, determining implied non-edges and {\\em dependent} edge sets remains an elusive, long-standing open problem.\n  One obstacle is to determine when implied non-edges can exist without non-trivial rigid induced subgraphs, i.e., {\\em nucleations}, and how to deal with them.\n  In this paper, we give general inductive construction schemes and proof "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4859","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"}