10,718 research outputs found
Higher signature Delaunay decompositions
A Delaunay decomposition is a cell decomposition in R^d for which each cell
is inscribed in a Euclidean ball which is empty of all other vertices. This
article introduces a generalization of the Delaunay decomposition in which the
Euclidean balls in the empty ball condition are replaced by other families of
regions bounded by certain quadratic hypersurfaces. This generalized notion is
adaptable to geometric contexts in which the natural space from which the point
set is sampled is not Euclidean, but rather some other flat semi-Riemannian
geometry, possibly with degenerate directions. We prove the existence and
uniqueness of the decomposition and discuss some of its basic properties. In
the case of dimension d = 2, we study the extent to which some of the
well-known optimality properties of the Euclidean Delaunay triangulation
generalize to the higher signature setting. In particular, we describe a higher
signature generalization of a well-known description of Delaunay decompositions
in terms of the intersection angles between the circumscribed circles.Comment: 25 pages, 6 figure
Characterizing the Delaunay decompositions of compact hyperbolic surfaces
Given a Delaunay decomposition of a compact hyperbolic surface, one may
record the topological data of the decomposition, together with the
intersection angles between the `empty disks' circumscribing the regions of the
decomposition. The main result of this paper is a characterization of when a
given topological decomposition and angle assignment can be realized as the
data of an actual Delaunay decomposition of a hyperbolic surface.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper12.abs.htm
On contractible edges in convex decompositions
Let be a convex decomposition of a set of points in
general position in the plane. If consists of more than one polygon, then
either contains a deletable edge or contains a contractible edge
Coloring decompositions of complete geometric graphs
A decomposition of a non-empty simple graph is a pair , such that
is a set of non-empty induced subgraphs of , and every edge of
belongs to exactly one subgraph in . The chromatic index of a
decomposition is the smallest number for which there exists a
-coloring of the elements of in such a way that: for every element of
all of its edges have the same color, and if two members of share at
least one vertex, then they have different colors. A long standing conjecture
of Erd\H{o}s-Faber-Lov\'asz states that every decomposition of the
complete graph satisfies . In this paper we work
with geometric graphs, and inspired by this formulation of the conjecture, we
introduce the concept of chromatic index of a decomposition of the complete
geometric graph. We present bounds for the chromatic index of several types of
decompositions when the vertices of the graph are in general position. We also
consider the particular case in which the vertices are in convex position and
present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure
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