Title:
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On structural and graph theoretic properties of higher order Delaunay graphs
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Author:
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Abellanas, Manuel; Bose, Prosenjit; García López de Lacalle, Jesús; Hurtado Díaz, Fernando Alfredo; Nicolás, Carlos M.; Ramos, Pedro A.
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Other authors:
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Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II; Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta |
Abstract:
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Given a set $\emph{P}$ of $\emph{n}$ points in the plane, the order-$\emph{k}$ Delaunay graph is a graph with vertex set $\emph{P}$ and an edge exists between two points p,q ∊ $\emph{P}$ when there is a circle through $\emph{p}$ and $\emph{q}$ with at most $\emph{k}$ other points of $\emph{P}$ in its interior. We provide upper and lower bounds on the number of edges in an order-$\emph{k}$ Delaunay graph. We study the
combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-$\emph{k}$ Delaunay graph is connected under the flip operation when $\emph{k}$ ≤ 1 but not necessarily connected for other values of $\emph{k}$. If $\emph{P}$ is in convex position then the order-$\emph{k}$ Delaunay graph is connected for all $\emph{k}$ ≥ 0.
We show that the order-$\emph{k}$ Gabriel graph, a subgraph of the order-$\emph{k}$ Delaunay graph, is
Hamiltonian for $\emph{k}$ ≥ 15. Finally, the order-$\emph{k}$ Delaunay graph can be used to effciently
solve a coloring problem with applications to frequency assignments in cellular networks. |
Subject(s):
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-Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs -Graph theory -Combinatorial analysis -Voronoi polygons -Triangulations -Grafs, Teoria de -Anàlisi combinatòria -Voronoi, Polígons de -Triangulació |
Rights:
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Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
Document type:
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Article - Published version Article |
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