Injective Split Systems

Abstract

AbstractA split system$$\mathcal S$$Son a finite setX,$$|X|\ge 3$$|X|≥3, is a set of bipartitions or splits ofXwhich contains all splits of the form$$\{x,X-\{x\}\}$${x,X-{x}},$$x \in X$$x∈X. To any such split system$$\mathcal S$$Swe can associate the Buneman graph$$\mathcal B(\mathcal S)$$B(S)which is essentially a median graph with leaf-setXthat displays the splits in$$\mathcal S$$S. In this paper, we consider properties of injective split systems, that is, split systems$$\mathcal S$$Swith the property that$${{\,\textrm{med}\,}}_{\mathcal B(\mathcal S)}(Y) \ne {{\,\textrm{med}\,}}_{\mathcal B(\mathcal S)}(Y')$$medB(S)(Y)≠medB(S)(Y′)for any 3-subsets$$Y,Y'$$Y,Y′inX, where$${{\,\textrm{med}\,}}_{\mathcal B(\mathcal S)}(Y)$$medB(S)(Y)denotes the median in$$\mathcal B(\mathcal S)$$B(S)of the three elements inYconsidered as leaves in$$\mathcal B(\mathcal S)$$B(S). In particular, we show that for any setXthere always exists an injective split system onX, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph$$\mathcal B(\mathcal S)$$B(S)needs to become in order for a split system$$\mathcal S$$SonXto be injective. We do this by introducing a quantity for |X| which we call the injective dimension for |X|, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map onXcan be represented using Buneman graphs.