For a given set of taxa, and a set of splits S on the taxa, usually together with a non-negative weighting, which may represent character changes distance, or may also have a more abstract interpretation, if the set of splits S is compatible, then it can be represented by an unrooted phylogenetic tree and each edge in the tree corresponds to exactly one of the splits. More generally, S can always be represented by a split network, which is an unrooted phylogenetic network with the property that every split in S is represented by an array of parallel edges in the network.
References
Further reading Huson, Daniel H.; Scornavacca, Celine (2011). "A survey of combinatorial methods for phylogenetic networks". Genome Biology and Evolution. 3: 23–35. doi:10.1093/gbe/evq077. PMC 3017387. PMID 21081312. Huson, Daniel H.; Rupp, Regula; Scornavacca, Celine (2011). Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press. ISBN 978-0521755962.