Jiayue Qi, Josef Schicho
A phylogenetic tree is a tree with a fixed set of leaves that has no vertices of degree two. In this paper, we axiomatically define four other discrete structures on the set of leaves. We prove that each of these structures is an equivalent representation of a phylogenetic tree.
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Unrooted phylogenetic networks are graphs used to represent evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all phylogenetic networks with at least five leaves can be uniquely reconstructed from their subnetworks obtained by deleting a single leaf and suppressing degree-2 vertices. Here, we show that this conjecture is false, by presenting a counter example for each possible number of leaves that is at least~4. Moreover, we show that the conjecture is still false when restricted to binary …
The maximum quartet distance between phylogenetic trees
该研究解决了1986年Bandelt和Dress的猜想，确定了二元系统发育树间四重态距离的最大值，通过共根平面化等方法完成证明，校准了比较系统发育树的基础度量尺度。
The Shapley value, a solution concept from cooperative game theory, has recently been considered for both unrooted and rooted phylogenetic trees. Here, we focus on the Shapley value of unrooted trees and first revisit the so-called split counts of a phylogenetic tree and the Shapley transformation matrix that allows for the calculation of the Shapley value from the edge lengths of a tree. We show that non-isomorphic trees may have permutation-equivalent Shapley transformation matrices and permutation-equivalent null spaces. This implies that estimating the split counts associated with a tree o…
Given a collection $τ$ of subsets of a finite set $X$, we say that $τ$ is {\em phylogenetically flexible} if, for any collection $R$ of rooted phylogenetic trees whose leaf sets comprise the collection $τ$, $R$ is compatible (i.e. there is a rooted phylogenetic $X$--tree that displays each tree in $R$). We show that $τ$ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being `slim'. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not $τ$ is slim. This `slim' condition reduces to a simpler ine…
Order-Dependent Dissimilarity Measures on Phylogenetic Trees
本文比较有根系统发育树的三种阶依赖向量表示度量与rSPR距离、杂交数等指标的关系，证明在特定排序最小化下杂交数与时间树-子杂交数分别与这些向量度量等价。
Driven by the need for better models that allow one to shed light into the question how life's diversity has evolved, phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does…
It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this "rogue taxon" effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a "rogue taxon" such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the …
Metropolis-Hastings Sampling of Phylogenetic Networks: Correcting for Symmetries
该研究针对系统发育网络的Metropolis-Hastings采样提出内部对称性校正方法，利用μ-向量简化自同构群计算，可加快采样速度，果园网络等无需校正。
We describe a kernel of size 9k-8 for the NP-hard problem of computing the Tree Bisection and Reconnect (TBR) distance k between two unrooted binary phylogenetic trees. We achieve this by extending the existing portfolio of reduction rules with three novel new reduction rules. Two of the rules are based on the idea of topologically transforming the trees in a distance-preserving way in order to guarantee execution of earlier reduction rules. The third rule extends the local neighbourhood approach introduced in (Kelk and Linz, Annals of Combinatorics 24(3), 2020) to more global structures, allo…
Given two phylogenetic trees with the $\{1, \ldots, n\}$ leaf-set the maximum agreement subtree problem asks what is the maximum size of the subset $A \subseteq \{1, \ldots, n\}$ such that the two trees are equivalent when restricted to $A$. The long-standing extremal version of this problem focuses on the smallest number of leaves, $\mathrm{mast}(n)$, on which any two (binary and unrooted) phylogenetic trees with $n$ leaves must agree. In this work we prove that this number grows asymptotically as $Θ(\log n)$; thus closing the enduring gap between the lower and upper asymptotic bounds on $\ma…
How many cherry-picking sequences are needed to reduce all subtrees of a phylogenetic tree?
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。
A Wild Sheep Chase Through an Orchard
本文研究无向果园网络，证明判定无向二元系统发生网络是否为果园是NP困难的，并引入新刻画以助未来正面结论。
Orthology and Near-Cographs in the Context of Phylogenetic Networks
本文针对系统发育网络下的直系同源图展开研究，提出1级可解释直系同源图的特征（原始子图为近补图），并设计线性时间算法识别该类图及构建对应1级网络，解决类网络进化场景下的直系同源图解释问题。
In this paper, we investigate a conjecture by von Haeseler concerning the Maximum Parsimony method for phylogenetic estimation, which was published by the Newton Institute in Cambridge on a list of open phylogenetic problems in 2007. This conjecture deals with the question whether Maximum Parsimony trees are hereditary. The conjecture suggests that a Maximum Parsimony tree for a particular (DNA) alignment necessarily has subtrees of all possible sizes which are most parsimonious for the corresponding subalignments. We answer the conjecture affirmatively for binary alignments on five taxa but a…
On the correctness of Maximum Parsimony for data with few substitutions in the NNI neighborhood of phylogenetic trees
本文推广了关于最大简约法正确性的已有结果，证明在特定条件下（如n≥12k），包含所有在树T上需要k次替换的二元性状的比对在T的NNI邻域内以T为唯一最简约树，部分肯定地回答了一个猜想，并构造了k约为n/2时NNI邻居间简约性反转的反例。
NOC NOC, who's there? Clustering systems of tree-child and normal networks
本文通过非重叠覆盖（NOC）性质刻画正规和树孩子网络的聚类系统，证明NOC等价于包含可见性，给出二次上界与多项式时间识别算法，并探讨了其在枚举、序理论和最小集合覆盖等方面的推论。
Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples
该研究通过并闭族的高度证明了Frankl并闭集猜想在高度≤4时成立，推导了高度≤5的空集反例的结构性质，排除了临界并覆盖数为5的情况，为高度5的反例提供了关键约束。
In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with girth at least four or five, respectively, has an induced forest on at least $n-\frac{2}{9}m$ or $n-\frac{1}{5}m$ vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such gra…
Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Q(G) and of its subposet of the G-symmetric partitions Q_G which was recently introduced by Hultman together with the complex of G-symmetric phylogenetic trees T_G. Hultman shows that T_G and Q_G are homotopy equivalent and Cohen-Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov shows that in fact T_G is subdivided by the order complex of Q_G. We introduce the c…
Network Representation and Modular Decomposition of Combinatorial Structures: A Galled-Tree Perspective
本文研究组合结构 strudigrams 的网络表示，提出用 galls 局部替换模分解树中的素顶点构建标记 galled-trees，定义并刻画了 galled-tree 可解释（GATEX）strudigrams，证明其识别与重建可在多项式时间内完成。
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