Injective Split Systems

Abstract

A split system mathcal S on a finite set X, |X|ge 3, is a set of bipartitions or splits of X which contains all splits of the form {x,X-{x}}, x in X. To any such split system mathcal S we can associate the Buneman graph mathcal B(mathcal S) which is essentially a median graph with leaf-set X that displays the splits in mathcal S. In this paper, we consider properties of injective split systems, that is, split systems mathcal S with the property that {{,textrm{med},}}_{mathcal B(mathcal S)}(Y) ne {{,textrm{med},}}_{mathcal B(mathcal S)}(Y') for any 3-subsets Y,Y' in X, where {{,textrm{med},}}_{mathcal B(mathcal S)}(Y) denotes the median in mathcal B(mathcal S) of the three elements in Y considered as leaves in mathcal B(mathcal S). In particular, we show that for any set X there always exists an injective split system on X, 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) needs to become in order for a split system mathcal S on X to 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 on X can be represented using Buneman graphs.