A decomposition theory for phylogenetic networks and incompatible characters.

Phylogenetic networks are models of evolution that go beyond trees, incorporating non-tree-like biological events such as recombination (or more generally reticulation), which occur either in a single species (meiotic recombination) or between species (reticulation due to lateral gene transfer and hybrid speciation). The central algorithmic problems are to reconstruct a plausible history of mutations and non-tree-like events, or to determine the minimum number of such events needed to derive a given set of binary sequences, allowing one mutation per site. Meiotic recombination, reticulation and recurrent mutation can cause conflict or incompatibility between pairs of sites (or characters) of the input. Previously, we used "conflict graphs" and "incompatibility graphs" to compute lower bounds on the minimum number of recombination nodes needed, and to efficiently solve constrained cases of the minimization problem. Those results exposed the structural and algorithmic importance of the non-trivial connected components of those two graphs. In this paper, we more fully develop the structural importance of non-trivial connected components of the incompatibility and conflict graphs, proving a general decomposition theorem (Gusfield and Bansal, 2005) for phylogenetic networks. The decomposition theorem depends only on the incompatibilities in the input sequences, and hence applies to many types of phylogenetic networks, and to any biological phenomena that causes pairwise incompatibilities. More generally, the proof of the decomposition theorem exposes a maximal embedded tree structure that exists in the network when the sequences cannot be derived on a perfect phylogenetic tree. This extends the theory of perfect phylogeny in a natural and important way. The proof is constructive and leads to a polynomial-time algorithm to find the unique underlying maximal tree structure. We next examine and fully solve the major open question from Gusfield and Bansal (2005): Is it true that for every input there must be a fully decomposed phylogenetic network that minimizes the number of recombination nodes used, over all phylogenetic networks for the input. We previously conjectured that the answer is yes. In this paper, we show that the answer in is no, both for the case that only single-crossover recombination is allowed, and also for the case that unbounded multiple-crossover recombination is allowed. The latter case also resolves a conjecture recently stated in (Huson and Klopper, 2007) in the context of reticulation networks. Although the conjecture from Gusfield and Bansal (2005) is disproved in general, we show that the answer to the conjecture is yes in several natural special cases, and establish necessary combinatorial structure that counterexamples to the conjecture must possess. We also show that counterexamples to the conjecture are rare (for the case of single-crossover recombination) in simulated data.