Separation by Convex Pseudo-Circles

N Chevallier (1) , A Fruchard (1) , D Schmitt (1) , J.-C Spehner (1)

1 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]

Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal n 0 + n 1 + n 2 + n 3 . This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. Buzaglo, Holzman, and Pinchasi showed that it is an upper bound for the number of subsets separable by (non necessarily convex) pseudo-circles. Actually, we first count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles, for a given k. We show that Lee's result on the number of k-subsets separable by true circles still holds for convex pseudo-circles. In particular, this means that the number of k-subsets of S separable by a maximal family of convex pseudo-circles is an invariant of S: It does not depend on the choice of the maximal family. To prove this result, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram, and whose vertices are the k-subsets of S separable by a maximal family of convex pseudo-circles. In order to count the number of vertices of this graph, we first show that it admits a planar realization which is a triangulation. It turns out (but is not detailed in the present paper) that these triangulations are the centroid triangulations Liu and Snoeyink conjectured to construct.

Mots clés: pseudo-circle; k-sets; centroid triangulation; VC-dimension

Domaines: Géométrie algorithmique [cs.CG]; Combinatoire [math.CO]

N Chevallier, A Fruchard, D Schmitt, J.-C Spehner. Separation by Convex Pseudo-Circles. 2015. ⟨hal-01108235⟩