Attaching leaves and picking cherries to characterise the hybridisation number for a set of phylogenies

Abstract

Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when |P| = 2. However, despite various attempts, characterising this number in terms of agreement forests for |P| > 2 remains elusive. In this paper, we characterise the minimum hybridisation number for when P is of arbitrary size and consists of not necessarily binary trees. Building on our previous work on cherry-picking sequences, we first establish a new characterisation to compute the minimum hybridisation number in the space of tree-child networks. Subsequently, we show how this characterisation extends to the space of all rooted phylogenetic networks. Moreover, we establish a particular hardness result that gives new insight into some of the limitations of agreement forests.

Combining Networks Using Cherry Picking Sequences

“…number that displays all input tree-child networks on the same set of taxa. We generalize the results presented in [14] (they considered inputs of trees while we consider inputs of networks) by showing how this problem relates to the more generalized problem of Network Hybridization and also to the Tree-child Weight problem. For the Tree-child Network Hybridization problem, it turns out that there is not always a solution for some given inputs; we also comment on when this is the case.…”

“…This problem can be viewed as the natural extension of the classic Hybridization problem for trees. Linz and Semple show that Hybridization can be solved by adding leaves in the right place to all input trees, and then solving Tree-child Hybridization [14]. This also holds for the network versions of these problems, as the solution to Network Hybridization can be made treechild by adding leaves, and all networks displayed by a tree-child network are tree-child networks as well (Fig.…”

“…By Lemma 8, S also reduces all displayed networks of N , and hence it reduces N . Furthermore, the weight of S is equal to the reticulation number of N by Lemma 3 from [13], (originally proved slightly less strong in [14]). Now suppose there exists a TCS S that reduces N .…”

“…2) inputs and output, we can use tree-child sequences to obtain an FPT algorithm. This FPT algorithm is an extension of the one introduced in [7] in which they considered tree inputs; the tree-child sequence approach was first introduced in [14]. We also comment briefly on how some measure of an optimal solution to the Network Hybridization problem can be characterized by solving this restricted problem.…”

The rigid hybrid number for two phylogenetic trees

“…2, Fig. 3) and represent a hypothetical evolutionary scenario tracing the evolution of a genomic region within four species for when collections of trees are displayed by special types of networks (Humphries et al 2013a;Linz and Semple 2019) and related algorithms/complexity results (Bordewich and Semple 2007a, b;Döcker et al 2019;Humphries et al 2013b;Huson and Linz 2016). However, all of these results rely on the fact that the networks display the set of trees in question.…”