{"paper":{"title":"A new graph parameter related to bounded rank positive semidefinite matrix completions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.OC","authors_text":"Antonios Varvitsiotis, Monique Laurent","submitted_at":"2012-04-03T17:10:31Z","abstract_excerpt":"The Gram dimension $\\gd(G)$ of a graph $G$ is the smallest integer $k\\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists). For any fixed $k$ the class of graphs satisfying $\\gd(G) \\le k$ is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is $K_{k+1}$ for $k\\le 3$ and that there are two mini"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0734","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"}