# Toughness and Delaunay triangulations

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DOI: 10.1007/BF02187810

- Cite this article as:
- Dillencourt, M.B. Discrete Comput Geom (1990) 5: 575. doi:10.1007/BF02187810

## Abstract

We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graph*G* is*1-tough* if for any set*P* of vertices,*c*(*G−P*)≤|*G*|, where*c*(*G−P*) is the number of components of the graph obtained by removing*P* and all attached edges from*G*, and |*G*| is the number of vertices in*G*. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulations*T* satisfy the following closely related property: for any set*P* of vertices the number of interior components of*T−P* is at most |*P*|−2, where an interior component of*T−P* is a component that contains no boundary vertex of*T*. 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.