Comparison of Galled Trees

Galled trees, directed acyclic graphs that model evolutionary histories with isolated hybridization events, have become very popular due to both their biological significance and the existence of polynomial time algorithms for their reconstruction. In this paper we establish to which extent several distance measures for the comparison of evolutionary networks are metrics for galled trees, and hence when they can be safely used to evaluate galled tree reconstruction methods.Comment: 36 page

The maximum agreement of two nested phylogenetic networks

Given a set N of phylogenetic networks, the maximum agreement phylogenetic subnetwork problem (MASN) asks for a subnetwork embedded in every Ni ∈ N with as many leaves as possible. MASN can be used to identify shared branching structure among phylogenetic networks or to measure their similarity. In this chapter, we prove that the general case of MASN is NP-hard already for two phylogenetic networks (in fact, even if one of the two input networks is a binary tree), but that the problem can be solved efficiently if each of the two input phylogenetic networks exhibits a nested structure. For this purpose, we introduce the concept of a nested phylogenetic network and study some of its underlying fundamental combinatorial properties. We first show that the total number of nodes |V (N) | in any nested phylogenetic network N with n leaves and nesting depth d is O(n(d + 1)). We then describe a simple algorithm for testing if a given phylogenetic network is nested, and if so, determining its nesting depth in O(|V (N) | · (d + 1)) time. Next, we present a polynomial-time algorithm for MASN for two nested phylogenetic networks N1,N2. Its running time is O(|V (N1) |