Injective Split Systems

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