Tree-based unrooted nonbinary phylogenetic networks

Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered 'tree-like' or not, which leads to the introduction of tree-based networks in the rooted, binary context. Tree-based networks are those networks which can be constructed by adding additional edges into a phylogenetic tree, called the base tree. Previous extensions have considered extending to the binary, unrooted case and the nonbinary, rooted case. In this paper, we extend tree-based networks to the context of unrooted, nonbinary networks in three ways, depending on the types of additional edges that are permitted. Further, we study fully tree-based networks which are phylogenetic networks in which every embedded tree is a base tree. We also extend this concept to unrooted, nonbinary, phylogenetic networks and classify the resulting networks. Finally, we derive some results on the colourability of tree-based networks, which can be useful to determine whether a network is tree-based.