Computing the Maximum Agreement of Phylogenetic Networks

We introduce the maximum agreement phylogenetic subnetwork problem (MASN) of finding a branching structure shared by a set of phylogenetic networks. We prove that the problem is NP-hard even if restricted to three phylogenetic networks and give an O(n²)-time algorithm for the special case of two level-1 phylogenetic networks, where n is the number of leaves in the input networks and where N is called a level-f phylogenetic network if every biconnected component in the underlying undirected graph contains at most f nodes having indegree 2 in N. Our algorithm can be extended to yield a polynomial-time algorithm for two level-f phylogenetic networks N1,N2 for any f which is upper-bounded by a constant; more precisely, its running time is O(|V (N1)|·|V (N2)|·4 f), where V (Ni) denotes the set of nodes of Ni