Graph-theoretical conditions for inscribability and Delaunay realizability

Abstract

We present new graph-theoretical conditions for inscribable polyhedra and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is inscribable. These results have several consequences: • All 4-connected polyhedra are inscribable. • All simplicial polyhedra in which aU vertex degrees are between 4 and 6, inclusive, are inscribable. • AU triangulations without chords or nonfacial triangles are realizable as Delaunay triangulations. 'The support of the Committee on Research of the University of California, Irvine is gratefully acknowledged. 'By Lemma 2.2 of this paper, the result in [7] implies that any pyramid with a triangulated base is inscribable.

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

“…The algorithm has been simplified later in [18,26]. Other connections between toughness, polyhedra of inscribable type, and L 2 -Delaunay graphs have been developed in [10]. For an arbitrary convex distance function Γ , the Γ -Delaunay realizability has not yet been studied.…”

Outerplanar Graphs and Delaunay Triangulations

“…The first proof that an outerplanar graph can be realized as a Delaunay triangulation relies on two elegant results due to Dillencourt and Smith [3] and to Rivin [4,5] Next, we need this result of Rivin [4,5]. To complete our first proof, we just need to show that:…”

“…We are not aware of them previously appearing in the literature. The first one is an easy consequence of Dillencourt and Smith's [3] criterion relating Hamiltonianicity and inscribability. The second one, which occupies most of this note, uses Rivin's [4] inscribability criterion and constructs an explicit "witness" of this inscribability, in the form of certain weights assigned to the edges of the graph.…”

The Three Dimensional Logic Engine

“…Our proof builds a 3D version of a "logic engine". The building blocks we designed are based on the structure of diamond; they may prove useful in extending the logic engine approach to obtain complexity results for other 3D layout and proximity problems studied previously in two dimensions (e.g., [3,4,6,7,[12][13][14]). …”