编码三级半有向系统发育网络的四元网络与五元网络
Encoding level-3 semi-directed phylogenetic networks by quarnets and quinnets
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中文总结 AI 辅助
本文证明三级半有向系统发育网络可由五元网络编码，并精确刻画四元网络的唯一反例障碍，其他三级网络均可由四元网络编码。
中文摘要 AI 辅助
系统发育网络将系统发育树推广为进化历史的模型，允许谱系既分化又融合。对于许多类型的遗传数据，此类网络的根位置无法恢复，因此只能推断出半有向网络：一种混合图，其中仅进入网状顶点的边是有向的。推断此类网络的常用策略是首先推断其在每组$k\geq 3$个叶子上的诱导子网络（称为$k$-网络），然后组装这些片段。这仅在$k$-网络确定网络时才能成功，此时称该网络由其$k$-网络编码。已知一级和二级半有向网络（其双连通分量分别包含至多一个或两个网状结构）由其$4$-网络（即四元网络）编码，而三级网络则不然。尽管如此，在本文中我们证明三级半有向网络由其$5$-网络（即五元网络）编码，并精确刻画了四元网络的局限性：我们证明一个先前报道的反例捕获了唯一的障碍，所有其他三级网络均由其四元网络编码。我们的证明基于对网络各个结构特征的编码结果集合，这些结果对任意级别的网络均成立，且具有独立意义。
英文摘要
Phylogenetic networks generalize phylogenetic trees as models of evolutionary history, allowing lineages to merge as well as to diverge. For many types of genetic data the root position of such a network cannot be recovered, so that only a semi-directed network can be inferred: a mixed graph in which only the edges entering a reticulation vertex are directed. A common strategy for inferring such a network is to first infer the subnetwork it induces on each set of $k\geq 3$ of its leaves, called a $k$-net, and then to assemble these pieces. This can only succeed if the $k$-nets determine the network, in which case that network is said to be encoded by its $k$-nets. Semi-directed networks of level-1 and 2, those whose biconnected components contain at most one, respectively two, reticulations, are known to be encoded by their $4$-nets, or quarnets, whereas level-3 networks are not. Even so, in this paper we show that level-3 semi-directed networks are encoded by their $5$-nets, or quinnets, and we characterize the limitation of quarnets exactly: we show that a single previously reported counterexample captures the only obstruction, every other level-3 network being encoded by its quarnets. Our proofs rest on a collection of encoding results for individual structural features of a network, which we establish for networks of arbitrary level and which are of independent interest.
发表机构
- Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）
- School of Computing Sciences, University of East Anglia（东英吉利大学计算科学学院）
机构由 AI 辅助整理，请以论文原文为准。
