Polynomial invariants for cactuses

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...

Level-2 networks from shortest and longest distances

Recently it was shown that a certain class of phylogenetic networks, called level-2 networks, cannot be reconstructed from their associated distance matrices. In this paper, we show that they can be reconstructed from their induced shortest and longest distance matrices. That is, if two level-2 networks induce the same shortest and longest...

Reconstructibility of unrooted level-k phylogenetic networks from distances

A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives...

Leaf-Reconstructibility of Phylogenetic Networks

An important problem in evolutionary biology is to reconstruct the evolutionary history of a set X of species. This history is often represented as a phylogenetic network, that is, a connected graph with leaves labelled by elements in X (for example, an evolutionary tree), which is usually also binary, i.e., all vertices have degree 1 or 3. A...