Recently there has been considerable interest in the problem of finding a phylogenetic network with a minimum number of reticulation vertices which displays a given set of phylogenetic trees, that is, a network with minimum hybrid number. Such networks are useful for representing the evolution of species whose genomes have undergone processes such as lateral gene transfer and recombination that cannot be represented appropriately by a phylogenetic tree. Even so, as was recently pointed out in the literature, insisting that a network displays the set of trees can be an overly restrictive assumption when modeling certain evolutionary phenomena such as incomplete lineage sorting. In this paper, we thus consider the less restrictive notion of rigidly displaying which we introduce and study here. More specifically, we characterize when two trees can be rigidly displayed by a certain type of phylogenetic network called a temporal tree-child network in terms of fork-picking sequences. These are sequences of special subconfigurations of the two trees related to the well-studied cherry-picking sequences. We also show that, in case it exists, the rigid hybrid number for two phylogenetic trees is given by a minimum weight fork-picking sequence for the trees. Finally, we consider the relationship between the rigid hybrid number and three closely related numbers; the weak, beaded, and temporal hybrid numbers. In particular, we show that these numbers can all be different even for a fixed pair of trees, and also present an infinite family of pairs of trees which demonstrates that the difference between the rigid hybrid number and the temporal-hybrid number for two phylogenetic trees on the same set of n leaves can grow at least linearly with n.

最近，人们对寻找具有最小数量网状顶点的系统发育网络的问题产生了相当大的兴趣，该网络显示给定的一组系统发育树，即具有最小杂合数的网络。这种网络可用于表示物种的进化，这些物种的基因组经历了横向基因转移和重组等过程，而这些过程无法通过系统发育树适当地表示。即便如此，正如最近文献中指出的那样，在对某些进化现象（例如不完整的谱系排序）进行建模时，坚持认为网络显示树的集合可能是一个过于严格的假设。因此，在本文中，我们考虑了我们在这里介绍和研究的限制较少的严格显示概念。更具体地说，我们描述了两棵树何时可以通过某种类型的系统发育网络（称为时间树子网络）在分叉序列方面严格显示。这些是两棵树的特殊子配置序列，与经过充分研究的樱桃采摘序列相关。我们还表明，如果存在的话，两个系统发育树的刚性杂种数由树的最小权重分叉序列给出。最后，我们考虑刚性混合数与三个密切相关的数之间的关系；弱数、串珠数和时间混合数。特别是，我们表明，即使对于一对固定的树，这些数字也可能不同，并且还呈现了无限的树对家族，这证明了两个系统发育树的刚性杂交数和时间杂交数之间的差异在同一组n片叶子上可以至少与n线性增长。