A normal network is uniquely determined by the set of phylogenetic trees that it displays. Given a set $\mathcal{P}$ of rooted binary phylogenetic trees, this paper presents a polynomial-time algorithm that reconstructs the unique binary normal network whose set of displayed binary trees is $\mathcal{P}$, if such a network exists. Additionally, we show that any two rooted phylogenetic trees can be displayed by a normal network and show that this result does not extend to more than two trees. This is in contrast to tree-child networks where it has been previously shown that any collection of rooted phylogenetic trees can be displayed by a tree-child network. Lastly, we introduce a type of cherry-picking sequence that characterises when a collection $\mathcal{P}$ of rooted phylogenetic trees can be displayed by a normal network and, further, characterise the minimum number of reticulations needed over all normal networks that display $\mathcal{P}$. We then exploit these sequences to show that, for all $n\ge 3$, there exist two rooted binary phylogenetic trees on $n$ leaves that can be displayed by a tree-child network with a single reticulation, but cannot be displayed by a normal network with less than $n-2$ reticulations.

Magnus Bordewich Simone Linz Charles Semple

Rooted phylogenetic networks are used by biologists to infer and represent complex evolutionary relationships between species that cannot be accurately explained by a phylogenetic tree. Tree-child networks are a particular class of rooted phylogenetic networks that has been extensively investigated in recent years. In this paper, we give a novel characterisation of a tree-child network $\mathcal{R}$ in terms of cherry-picking sequences that are sequences on the leaves of $\mathcal{R}$ and reduce it to a single vertex by repeatedly applying one of two reductions to its leaves. We show that our characterisation extends to unrooted tree-child networks which are mostly unexplored in the literature and, in turn, also offers a new approach to settling the computational complexity of deciding if an unrooted phylogenetic network can be oriented as a rooted tree-child network.

Janosch Döcker Simone Linz

Agreement forests continue to play a central role in the comparison of phylogenetic trees since their introduction more than 25 years ago. More specifically, they are used to characterise several distances that are based on tree rearrangement operations and related quantifiers of dissimilarity between phylogenetic trees. In addition, the concept of agreement forests continues to underlie most advancements in the development of algorithms that exactly compute the aforementioned measures. In this paper, we introduce agreement digraphs, a concept that generalises agreement forests for two phylogenetic trees to two phylogenetic networks. Analogous to the way in which agreement forests compute the subtree prune and regraft distance between two phylogenetic trees but inherently more complex, we then use agreement digraphs to bound the subnet prune and regraft distance between two tree-child networks from above and below and show that our bounds are tight.

Steven Kelk Simone Linz Charles Semple

Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most 15k-9 taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here we propose five new reduction rules and show that these further reduce the bound to 11k-9. The new rules combine the ``unrooted generator'' approach introduced in [Kelk and Linz 2018] with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.

Steven Kelk Simone Linz

The search space of rooted phylogenetic trees is vast and a major research focus of recent decades has been the development of algorithms to effectively navigate this space. However this space is tiny when compared with the space of rooted phylogenetic networks, and navigating this enlarged space remains a poorly understood problem. This, and the difficulty of biologically interpreting such networks, obstructs adoption of networks as tools for modelling reticulation. Here, we argue that the superimposition of biologically motivated constraints, via an SQL-style language, can both stimulate use of network software by biologists and potentially significantly prune the search space.

Steven Kelk Simone Linz David A. Morrison

We consider canonical systems and investigate the Szeg\H{o} class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the canonical system. This sum rule provides a spectral theory gem in the sense proposed by Barry Simon.

David Damanik Benjamin Eichinger Peter Yuditskii

Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem of deciding if two phylogenetic networks embed the same sets of phylogenetic trees is computationally hard, \blue{in particular, we showed it to be $\Pi^P_2$-complete}. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consists of a normal network and a tree-child network. The running time of the algorithm is quadratic in the size of the leaf sets.

Janosch Doecker Simone Linz Charles Semple

Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network $N$ embeds as the display set of $N$, several questions in the context of the display set of $N$ have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given phylogenetic tree is contained in the display set of a given network. The focus of this paper are two questions that naturally arise in comparing the display sets of two phylogenetic networks. First, we analyze the problem of deciding if the display sets of two phylogenetic networks have a tree in common. Surprisingly, this problem turns out to be NP-complete even for two temporal normal networks. Second, we investigate the question of whether or not the display sets of two phylogenetic networks are equal. While we recently showed that this problem is polynomial-time solvable for a normal and a tree-child network, it is computationally hard in the general case. In establishing hardness, we show that the problem is contained in the second level of the polynomial-time hierarchy. Specifically, it is $\Pi_2^P$-complete. Along the way, we show that two other problems are also $\Pi_2^P$-complete, one of which being a generalization of Tree-Containment.

Janosch Döcker Simone Linz Charles Semple

In the study of rooted phylogenetic networks, analyzing the set of rooted phylogenetic trees that are embedded in such a network is a recurring task. From an algorithmic viewpoint, this analysis almost always requires an exhaustive search of a particular multiset $S$ of rooted phylogenetic trees that are embedded in a rooted phylogenetic network $\mathcal{N}$. Since the size of $S$ is exponential in the number of reticulations of $\mathcal{N}$, it is consequently of interest to keep this number as small as possible but without loosing any element of $S$. In this paper, we take a first step towards this goal by introducing the notion of a non-essential arc of $\mathcal{N}$, which is an arc whose deletion from $\mathcal{N}$ results in a rooted phylogenetic network $\mathcal{N}'$ such that the sets of rooted phylogenetic trees that are embedded in $\mathcal{N}$ and $\mathcal{N}'$ are the same. We investigate the popular class of tree-child networks and characterize which arcs are non-essential. This characterization is based on a family of directed graphs. Using this novel characterization, we show that identifying and deleting all non-essential arcs in a tree-child network takes time that is cubic in the number of leaves of the network. Moreover, we show that deciding if a given arc of an arbitrary phylogenetic network is non-essential is $\Pi_2^P$-complete.

Simone Linz Charles Semple

In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2^k, where k is the number of vertices with in-degree two in N, then G has a Hamiltonian cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.

Janosch Döcker Simone Linz Charles Semple