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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) < I GI, where c(G -P) is the number of components of the graph obtained by removing P and all attached edges from G, and I GI 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 thedoi:10.1007/bf02187810 fatcat:twkcz73tezdmxnygspfhdrdc6u