Obtaining splits from cut sets of tight spans

To any metric DD on a finite set XX, one can associate a metric space T(D)T(D) known as its tight span  . Properties of T(D)T(D) often reveal salient properties of DD. For example, cut sets of T(D)T(D), i.e., subsets of T(D)T(D) whose removal disconnect T(D)T(D), can help to identify clusters suggested by DD and indicate how T(D)T(D) (and hence DD) may be decomposed into simpler components. Given a bipartition or split SS of XX, we introduce in this paper a real-valued index ε(D,S)ε(D,S) that comes about by considering cut sets of T(D)T(D). We also show that this index is intimately related to another, more easily computable index δ(D,S)δ(D,S) whose definition does not directly depend on T(D)T(D). In addition, we provide an illustration for how these two new indices could help to extend and complement current distance-based methods for phylogenetic network construction such as split decomposition and NeighborNet.