Maximum agreement forests have been used as a measure of dissimilarity of two or more phylogenetic trees on a given set of taxa. An agreement forest is a set of trees that can be obtained from each of the input trees by deleting edges and suppressing degree-2 vertices. A maximum agreement forest is such a forest with the minimum number of components. We present a simple 4-approximation algorithm for computing a maximum agreement forest of multiple unrooted binary trees. This algorithm applies LP rounding to an extension of a recent ILP formulation of the maximum agreement forest problem on two trees by Van Wersch et al. [13] . We achieve the same approximation ratio as the algorithm by Chen et al. [3] , but our algorithm is extremely simple. We also prove that no algorithm based on the ILP formulation by Van Wersch et al. can achieve an approximation ratio of 4 − ε , for any ε > 0 , even on two trees. To this end, we prove that the integrality gap of the ILP approaches 4 as the size of the two input trees grows.

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m ≥ 2 . Moreover, for binary networks the problem remains NP-complete when m ≥ 3 but becomes polynomial-time solvable for m = 2 . We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based in terms of certain m -colorings. • Proper forest-based networks model evolutionary processes such as introgression. • We consider problem (P): Is a given m-rooted network N proper forest-based? • We show (P) can be solved in polynomial time if N is 2-rooted, binary tree-child. • We show (P) is NP-complete if N is m-rooted, binary tree-child with m ≥ 3. • We give an FPT algorithm for (P) in case every vertex in N has indegree at most 2.

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial F that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial Q that is a complete invariant for the class of rooted cactuses that has 6 variables whose degree can be bounded linearly in terms of the size of the rooted cactus. We also explain how to extend the Q polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.

Evolutionary histories for species that cross with one another or exchange genetic material can be represented by leaf-labelled, directed graphs called phylogenetic networks. A major challenge in the burgeoning area of phylogenetic networks is to develop algorithms for building such networks by amalgamating small networks into a single large network. The level of a phylogenetic network is a measure of its deviation from being a tree; the higher the level of a network, the less treelike it becomes. Various algorithms have been developed for building level-1 networks from small networks. However, level-1 networks may not be able to capture the complexity of some data sets. In this paper, we present a polynomial-time algorithm for constructing a rooted binary level-2 phylogenetic network from a collection of 3-leaf networks or trinets. Moreover, we prove that the algorithm will correctly reconstruct such a network if it is given all of the trinets in the network as input. The algorithm runs in time O(t⋅n+n4) with t the number of input trinets and n the number of leaves. We also show that there is a fundamental obstruction to constructing level-3 networks from trinets, and so new approaches will need to be developed for constructing level-3 and higher level-networks.