Separation index of graphs and stacked 2-spheres

In 1987, Kalai proved that stacked spheres of dimension d≥3d≥3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the separation index. Namely, we show that the separation index of a triangulated 2-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all n-vertex triangulated 2-spheres, the separation index is minimised by some n  -vertex flag sphere for n≥6n≥6.Furthermore, we apply this characterisation of stacked 2-spheres to settle the outstanding 3-dimensional case of the Lutz–Sulanke–Swartz conjecture that “tight-neighbourly triangulated manifolds are tight”. For dimension d≥4d≥4, the conjecture has already been proved by Effenberger following a result of Novik and Swartz.