Trinets encode tree-child and level-2 phylogenetic networks

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that level-1 phylogenetic networks are encoded by their trinets, and also conjectured that all "recoverable" rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets

Placing problems from phylogenetics and (quantified) propositional logic in the polynomial hierarchy

In this thesis, we consider the complexity of decision problems from two different areas of research and place them in the polynomial hierarchy: phylogenetics and (quantified) propositional logic. In phylogenetics, researchers study the evolutionary relationships between species. The evolution of a particular gene can often be represented by a single phylogenetic tree. However, in order to model non-tree-like events on a species level such as hybridization and lateral gene transfer, phylogenetic networks are used. They can be considered as a structure that embeds a whole set of phylogenetic trees which is called the display set of the network. There are many interesting questions revolving around display sets and one is often interested in the computational complexity of the considered problems for particular classes of networks. In this thesis, we present our results for different questions related to the display sets of two networks and place the corresponding decision problems in the polynomial hierarchy. Another interesting question concerns the reconstruction of networks: given a set T of phylogenetic trees, can we construct a phylogenetic network with certain properties that embeds all trees in T? For a class of networks that satisfies certain temporal properties, Humphries et al. (2013) established a characterization for when this is possible based on the existence of a particular structure for T, a so-called cherry-picking sequence. We obtain several complexity results for the existence of such a sequence: Deciding the existence of a cherry-picking sequence turns out to be NP-complete for each non-trivial number (i.e., at least two) of given trees. Thereby, we settle the open question stated by Humphries et al. (2013) on the complexity for the case |T| = 2. On the positive side, we identify a special case that we place in the complexity class P by exploring connections to automata theory. Regarding propositional logic, we present our complexity results for the classical satisfiability problem (and variants resp. quantified generalizations thereof) and place the considered variants in the polynomial hierarchy. A common theme is to consider bounded variable appearances in combination with other restrictions such as monotonicity of the clauses or planarity of the incidence graph. This research was inspired by the conjecture that Monotone 3-SAT remains NP-complete if each variable appears at most five times which was stated in the scribe notes of a lecture held by Erik Demaine; we confirm this conjecture in an even more restricted setting where each variable appears exactly four times