Toughness and Delaunay triangulations

Abstract

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) < 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 the following closely related property: for any set P of vertices the number of interior components of T-P is at most IPI -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.

Blocking Delaunay triangulations

“…It is known that every Delaunay triangulation contains a perfect matching of its vertices [6]. Consider such a perfect matching M , and an independent set I .…”

Approximate proximity drawings

“…Since Delaunay triangulations are among the most studied graphs in computational geometry, we start by investigating the relationship between (ε 1 , ε 2 )-Delaunay drawings and Delaunay drawings. In the proof of the next result we will use the following result of Dillencourt [11]: Let Γ be a Delaunay drawing with possibly degenerate vertex set V . Add, if necessary, edges to Γ to obtain a triangulation of V .…”

Maximum Matchings and Minimum Blocking Sets in $$\varTheta _6$$ -Graphs

“…To obtain a lower bound on the size of a maximum matching it suffices, by Theorem 3, to find an upper bound on odd(G \ S) − |S| that holds for any S. We will use this approach in our proofs of Theorems 1 and 2. In fact, as in Dillencourt's proof [21] that Delaunay graphs have perfect matchings we will find an upper bound on comp(G \ S) − |S| that holds for any S, i.e., we establish a bound on the toughness of the graph [9].…”

“…One of the many beautiful properties of Delaunay triangulations is that they always contain a (near-)perfect matching, that is, at most one vertex is unmatched, as proved by Dillencourt [21]. This is one example of a structural property of a so-called proximity graph.…”

“…Lemma 3 (Dillencourt [21], Lemma 3.4). Let G be a triangulated planar graph and let S be a subset of vertices of G. Then every face of G[S] contains at most one component of G \ S.…”