Toughness and delaunay triangulations

Abstract

We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. Agraph with set of sites S is 1-tough if for any set P ⊆ S, c(S - P) ≤ \S\, where c(5 - P) is the number of components of the subgraph induced by the complement of P and |P| is the number of sites in P. We also showthat, under the same conditions, the number of interior components of S - P is at most \P\ - 2. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunaytriangulations. We give examples to show that these bounds can be attained, and we state and prove severalcorollaries. In particular, we show that maximal planar graphs inscribable in a sphere are 1-tough.