图的系统发育秩
The phylogenetic rank of a graph
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中文总结 AI 辅助
本文研究图的系统发育秩，提出贪心与精确算法计算之，构建6、7顶点图数据库，反驳遗传性等猜想，证明次可加性并分类秩1图。
中文摘要 AI 辅助
Pachter-Sturmfels 图的系统发育秩是嵌入图 G 所需度量树的最小数量，使得嵌入是等距的。这里，G 的所有边长度均为 1，度量树的乘积赋予上确界范数。我们开发了计算系统发育秩的贪心算法和精确算法。利用我们的算法，我们构建了一个包含所有 6 顶点和 7 顶点图的系统发育秩数据库。特别地，我们展示了反例，证明系统发育秩不是遗传的，不受 $\lceil \frac{n}{2} \rceil$ 的界限约束，并且 Pachter 和 Sturmfels 的广义四点猜想不成立。此外，我们证明了系统发育秩在 1-和以及某些 2-顶点-和下具有次可加性，并且它平凡地有上界 n-1，其中 n 表示顶点数。我们还提供了系统发育秩为 1 的图的完整分类，并构造了几个系统发育秩为 $\lceil \frac{n}{2} \rceil$ 的无限族。
英文摘要
The Pachter-Sturmfels phylogenetic rank of a graph G is the minimal number of metric trees needed to embed G isometrically. Here, all edges of G are of length one and the product of metric trees is endowed with the supremum norm. We develop both a greedy and an exact algorithm for computing phylogenetic ranks. Using our algorithms, we construct a database of phylogenetic ranks which includes all graphs on 6 and 7 vertices. In particular, we exhibit examples disproving that the phylogenetic rank is hereditary, bounded by $\lceil \frac{n}{2} \rceil$, and a generalised 4-point conjecture by Pachter and Sturmfels. In addition, we show that the phylogenetic rank is subadditive under 1-sums and certain 2-vertex-sums, and that it is trivially upper bounded by n-1, where n denotes the number of vertices. We also provide a complete classification of graphs with phylogenetic rank 1 and construct several infinite families with phylogenetic rank $\lceil \frac{n}{2} \rceil$.
发表机构
- Durham University（杜伦大学）
机构由 AI 辅助整理，请以论文原文为准。
