Toughness and Delaunay triangulations

Abstract

We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graphG is1-tough if for any setP of vertices,c(G−P)≤|G|, wherec(G−P) is the number of components of the graph obtained by removingP and all attached edges fromG, and |G| is the number of vertices inG. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulationsT satisfy the following closely related property: for any setP of vertices the number of interior components ofT−P is at most |P|−2, where an interior component ofT−P is a component that contains no boundary vertex ofT. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.