{"paper":{"title":"A universal tree-based network with the minimum number of reticulations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Charles Semple, Magnus Bordewich","submitted_at":"2017-07-26T02:17:01Z","abstract_excerpt":"A tree-based network $\\mathcal N$ on $X$ is universal if every rooted binary phylogenetic $X$-tree is a base tree for $\\mathcal N$. Hayamizu and, independently, Zhang constructively showed that, for all positive integers $n$, there exists an universal tree-based network on $n$ leaves. For all $n$, Hayamizu's construction contains $\\Theta(n!)$ reticulations, while Zhang's construction contains $\\Theta(n^2)$ reticulations. A simple counting argument shows that an universal tree-based network has $\\Omega(n\\log n)$ reticulations. With this in mind, Hayamizu as well as Steel posed the problem of de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.08274","kind":"arxiv","version":2},"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"}