Fumihiko Takeuchi

Tight degree bounds for pseudo-triangulations of points

We show that every set of n points in general position has a minimum pseudo-triangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudo-triangulation has at most four vertices). Both degree bounds are tight. Minimum pseudo-triangulations realizing these bounds (individually but not jointly) can be constructed in O(n logn) time.

A Branch-and-Cut Approach for Minimum Weight Triangulation

This paper considers the problem of computing a minimum weight triangulation of n points in the plane, which has been intensively studied in recent years in computational geometry. This paper investigates a branch-and-cut approach for minimum weight triangulations. The problem can be formulated as finding a minimum-weight maximal independent set of intersection graphs of edges. In combinatorial optimization, there are known many cuts for the independent set problem, and we further use a cut induced by geometric properties of triangulations. Combining this branch-and-cut approach with the β-skeleton method, the moderate-size problem could be solved efficiently in our computational experiments. Polyhedral characterizations of the proposed cut and applications of another old skeletal approach in mathematical programming as the independent set problem are also touched upon.

Incremental construction properties in dimension two: shellability, extendable shellability and vertex decomposability

We give new examples of shellable, but not extendably shellable two-dimensional simplicial complexes. They include minimal examples that are smaller than those previously known. We also give new examples of shellable, but not vertex decomposable two-dimensional simplicial complexes, including extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability implies the other. We found these examples by enumerating shellable two-dimensional simplicial complexes that are not pseudomanifolds.

Voronoi diagrams by divergences with additive weights

We introduce the Voronoi diagram by the divergence determincd by a convex function with additive weights. This class of Voronoi diagrams includes the Euclidean case and further the Voronoi diagram for normal distributions in a statistically meaningful setting. With the additive weights, the Voronoi diagram for circles is also included, These Voronoi diagrams can be handled in a unified way via appropriate potential functions and its tangent hypcrsurfaces. 1 Divcrgcnco Voronoi diagram In this acction, we define the Voronoi diagram by the divergence determined by a given convex function. The following preliminary results in sections 1.1 and 1.2 may be found in [9] and [l, 21, respectively. 1.1 Conjugacy of convex functions Let S be an open convex set in Rd. Let

Extremal Properties for Dissections of Convex 3-Polytopes

J be a twice differentiable and strictly convex function on S which diverges to &co at the infinity/boundary. Define ai

Nonregular Triangulations, View Graphs of Triangulations, and Linear Programming Duality

= a<+(0) for 0 = [O’] E S’ by and denote [B&l in Rd by &,!J. Define S+ c Rd by s

Enumerating Triangulations for Products of Two Simplices and for Arbitrary Configurations of Points

= {&b(e) ] e E S} Lemma 1 When 111 is twice differentiable and strictly convex, S” is an open convex set. Define a function 4: SG + R by for 1 E S

ENUMERATING TRIANGULATIONS IN GENERAL DIMENSIONS

, The supremum is attained for 8 with a

Enumerating triangulations for products of two simplices and for arbitrary configurations of points

= 7, and so define v(0) by 71(e) = w(e). ‘Tbc tanaor notation is used. Pcmksion to m&e digits1 or hard copies of all or part of this work for psrconal or clnmoom WC is gnu&d without fee provided that copies nro not mnde or dislribukd for profit or wmmercial advantage snd th3t copies bear Ohio notice end the full cifetion on the fti page. To copy oU~crv~i~, to republish, to post on servers or to redistribute to Iii require3 prior spccilic pcmksion andlor s fee. ~CCI 98 Minneapolis Minnesota USA Copyright ACM 1998 0-89791.973-4/98/6...

Incremental construction properties in dimension two-- shellability, extendable shellability and vertex decomposability.

5.00 Then, Lemma 2 (p is a twice diflerentiable and strictly convex function on 9. I