{"paper":{"title":"Some results on multithreshold graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Gregory J. Puleo","submitted_at":"2019-04-30T20:51:54Z","abstract_excerpt":"Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with thresholds $\\theta_1 , \\ldots, \\theta_k$ (strictly increasing) if one can assign real numbers $(r_v)_{v \\in V(G)}$, called ranks, such that for every pair of vertices $v,w$, we have $vw \\in E(G)$ if and only if the inequality $\\theta_i \\leq r_v + r_w$ holds for an odd number of indices $i$. When $k=1$ or $k=2$, the precise choice of thresholds $\\theta_1, \\ldots, \\theta_k$ does not matter, as a suitable transformation of the ranks transforms a representation with one choice of thresholds into a representation with any othe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.00099","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"}