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On structural and graph theoretic properties of higher order Delaunay graphs

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10.1142/S0218195909003143
 
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hdl:2117/9615

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Abellanas, Manuel
Bose, Prosenjit
García López de Lacalle, Jesús
Hurtado Díaz, Fernando AlfredoMés informacióMés informació
Nicolás, Carlos M.
Ramos, Pedro A.
Document typeArticle
Defense date2009-12
Rights accessOpen Access
Attribution-NonCommercial-NoDerivs 3.0 Spain
This work is protected by the corresponding intellectual and industrial property rights. Except where otherwise noted, its contents are licensed under a Creative Commons license : Attribution-NonCommercial-NoDerivs 3.0 Spain
Abstract
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.
CitationAbellanas, M. [et al.]. On structural and graph theoretic properties of higher order Delaunay graphs. "International journal of computational geometry and applications", Desembre 2009, vol. 19, núm. 6, p. 595-615. 
URIhttp://hdl.handle.net/2117/9615
DOI10.1142/S0218195909003143
ISSN0218-1959
Publisher versionhttp://www2.uah.es/pramos/Papers/k-dg-ijcga.pdf
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  • DCCG - Grup de recerca en geometria computacional, combinatoria i discreta - Articles de revista [85]
  • Departament de Matemàtiques - Articles de revista [3.473]
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