Discrete & Computational Geometry
, Volume 5, Issue 6, pp 575601
First online:
Toughness and Delaunay triangulations
 Michael B. DillencourtAffiliated withCenter for Automation Research and Institute of Advanced Computer Studies, University of Maryland
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We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1toughness. A graphG is1tough 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 1tough, 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 1tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.
 Title
 Toughness and Delaunay triangulations
 Journal

Discrete & Computational Geometry
Volume 5, Issue 6 , pp 575601
 Cover Date
 199012
 DOI
 10.1007/BF02187810
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 Springer New York
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 Authors

 Michael B. Dillencourt ^{(1)}
 Author Affiliations

 1. Center for Automation Research and Institute of Advanced Computer Studies, University of Maryland, 20742, College Park, MD, USA