Obtaining splits from cut sets of tight spans

To any metric D on a finite set X, one can associate a metric space T (D) known as its tight span. Properties of T (D) often reveal salient properties of D. For example, cut sets of T(D), i.e., subsets of T (D) whose removal disconnect T (D), can help to identify clusters suggested by D and indicate how T (D) (and hence D) may be decomposed into simpler components. Given a bipartition or splits of X, we introduce in this paper a real-valued index epsilon((D,S)) that comes about by considering cut sets of T (D). We also show that this index is intimately related to another, more easily computable index delta((D,S)) whose definition does not directly depend on 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. (C) 2013 Elsevier B.V. All rights reserved.