{"paper":{"title":"Intersection graphs of segments and $\\exists\\mathbb{R}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Jiri Matousek","submitted_at":"2014-06-10T17:20:49Z","abstract_excerpt":"A graph $G$ with vertex set $\\{v_1,v_2,\\ldots,v_n\\}$ is an intersection graph of segments if there are segments $s_1,\\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\\{v_i,v_j\\}$ is an edge of~$G$. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments.\n  It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by $\\exists\\mathbb{R}$. This class consists of problems that can be reduced, in polynomial time,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.2636","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"}