AI 中文总结
本文研究果园系统发育网络的覆盖数问题，通过引入子集连通性等概念，明确不同类型树的覆盖数计算方法，证明长度为组合数的单个序列可约化树的所有子树。
AI 中文摘要
系统发育网络是表示物种进化历史的图。最近，可通过 cherry-picking 序列约化的果园系统发育网络类因其计算和生物学特性受到关注。本文研究果园及其 cherry-picking 序列的一个基本问题，即覆盖数问题：给定一个果园网络 $N$，需要多少个 cherry-picking 序列才能约化 $N$ 的所有子网络？我们从考虑树的该问题入手，证明可通过类似但更精细的存活覆盖数概念，递归计算二叉树的覆盖数；同时给出非二叉树存活覆盖数的递归公式，但非二叉树覆盖数的计算似乎更具挑战性，对此我们证明星型树（根与所有叶节点相邻）的覆盖数等价于本文引入的子集连通性问题。最后，我们证明若对序列长度无限制，长度为 $\inom{n}{2}$ 的单个最小长度序列足以约化含 $n$ 个叶节点的树的所有子树。
英文摘要
Phylogenetic networks are graphs that represent the evolutionary history of species. Recently, the class of orchard phylogenetic networks, which can be reduced by so-called cherry-picking sequences, has gained attention for its computational and biological aspects. In this paper, we study a fundamental question on orchards and their cherry-picking sequences by considering the CoveringNumber problem: given an orchard network $N$, how many cherry-picking sequences are needed to reduce all subnetworks of $N$? We initiate this study by considering the problem for trees. We then show that the covering number can be computed for binary trees recursively using a similar but more fine-grained notion of survival covering number. We also give a recursive formula for the survival covering number of non-binary trees. However, computing the covering number for non-binary trees appears to be considerably more challenging. For this case, we show that the covering number of star trees (whose root is adjacent to all leaves) is equivalent to the so-called SubsetConnectivity problem, which we introduce in this paper. Finally, we show that if there is no restriction on the sequence length, a single sequence of minimum length $\binom{n}{2}$ suffices to reduce all subtrees of a tree on $n$ leaves.
Comments21 pages, 7 figures
