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Connections between thetagraphs, Delaunay triangulations, and orthogonal surfaces
 In Proceedings of the 36th International Conference on Graph Theoretic Concepts in Computer Science (WG 2010
, 2010
"... Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulard ..."
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Cited by 5 (2 self)
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Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulardistance Delaunay triangulations, introduced by Chew, have been shown to be plane 2spanners of the 2D Euclidean complete graph, i.e., the distance in the TDDelaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6graph defined in the 2D Euclidean space, namely the halfΘ6graph, composed of the evencone edges of the Θ6graph. Our main contribution is to show that these graphs are exactly the TDDelaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish: – Every Θ6graph is the union of two spanning TDDelaunay graphs. In particular, Θ6graphs are 2spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6graphs are tspanners for some constant t, and Θ7graphs were only known to be tspanners for t ≈ 7.562. – Every plane triangulation is TDDelaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TDDelaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.
Adjacency posets of planar graphs
 DISCRETE MATH
"... In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show t ..."
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Cited by 4 (3 self)
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In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertexface poset of such a graph.
Orthogonal Surfaces and their CPorders
, 2007
"... Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with c ..."
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Cited by 2 (2 self)
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Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and 3polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which nongeneric orthogonal surfaces have a polytopal structure. We review the state of knowledge of the 3dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cporders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cporders can lack key properties of face lattices. We investigate extra requirements which may help to have cporders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.
The Order Dimension of Planar Maps Revisited
"... Abstract. Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the BrightwellTrotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and fac ..."
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Abstract. Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the BrightwellTrotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyderpaths and Schnyderregions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(PM)) ≤ 4. This may be the first result in the area that is obtained without using the tools introduced by Schnyder.
DOI: 10.1007/9783642169267_25 Connections between ThetaGraphs, Delaunay Triangulations, and Orthogonal Surfaces ⋆
, 2010
"... Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulard ..."
Abstract
 Add to MetaCart
Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulardistance Delaunay triangulations, introduced by Chew, have been shown to be plane 2spanners of the 2D Euclidean complete graph, i.e., the distance in the TDDelaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6graph defined in the 2D Euclidean space, namely the halfΘ6graph, composed of the evencone
Hubert de FRAYSSEIX..................
, 2010
"... pour obtenir le grade de Docteur de l ’ École Polytechnique ..."