Results 1  10
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37
Improved Bounds for Planar kSets and Related Problems
, 1998
"... We prove an O.n.k C 1/1=3 / upper bound for planar ksets. This is the first considerable improvement on this bound after its early solution approximately 27 years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in the arrangement o ..."
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Cited by 115 (1 self)
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We prove an O.n.k C 1/1=3 / upper bound for planar ksets. This is the first considerable improvement on this bound after its early solution approximately 27 years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in the arrangement of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general.
Results on kSets and jFacets via Continuous Motion
"... Let be a set of points in in general position, i.e., no points on a commonflat,. Aset of is a set of points in that can be separated from by a hyperplane. Afacet of is an orientedsimplex spanned by points in which has exactly points from on the positive side of its affine hull. If is a planar po ..."
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Cited by 37 (10 self)
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Let be a set of points in in general position, i.e., no points on a commonflat,. Aset of is a set of points in that can be separated from by a hyperplane. Afacet of is an orientedsimplex spanned by points in which has exactly points from on the positive side of its affine hull. If is a planar point set and is even, a halving edge is an undirected edge between two points, such that the connecting line has the same number of points on either side. The number! "$ # ofsets is twice the number of halving edges. Inspired by Dey’s recent proof of a new bound on the number ofsets we show that
The maximum number of edges in quasiplanar graphs
"... A topological graph is quasiplanar, if it does not contain three pairwise crossing edges. Agarwal et al. [2] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) ..."
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Cited by 23 (4 self)
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A topological graph is quasiplanar, if it does not contain three pairwise crossing edges. Agarwal et al. [2] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasiplanar topological graph (i.e., every pair of edges have at most one point in common) is 6.5n − O(1), thereby exhibiting that nonsimple quasiplanar graphs may have many more edges than simple ones.
The SzemerédiTrotter theorem in the complex plane
, 305
"... This paper generalizes of the SzemerédiTrotter theorem to the complex plane. Szemerédi and Trotter proved that the number of pointline incidences of n points and e lines in the real Euclidean plane is O(n 2/3 e 2/3 + n + e). This bound is tight. Although several short proofs were found to this the ..."
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Cited by 22 (0 self)
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This paper generalizes of the SzemerédiTrotter theorem to the complex plane. Szemerédi and Trotter proved that the number of pointline incidences of n points and e lines in the real Euclidean plane is O(n 2/3 e 2/3 + n + e). This bound is tight. Although several short proofs were found to this theorem [14, 12], and many multidimensional generalizations were given, no tight bound has been known so far for incidences in higher dimensions. We extend the methods of Szemerédi and Trotter and prove that the number of pointline incidences of n points and e complex lines in the complex plane�2 is O(n
Fáry’s Theorem for 1Planar Graphs
"... Abstract. Fáry’s theorem states that every plane graph can be drawn as a straightline drawing. A plane graph is a graph embedded in a plane without edge crossings. In this paper, we extend Fáry’s theorem to nonplanar graphs. More specifically, we study the problem of drawing 1plane graphs with str ..."
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Cited by 15 (4 self)
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Abstract. Fáry’s theorem states that every plane graph can be drawn as a straightline drawing. A plane graph is a graph embedded in a plane without edge crossings. In this paper, we extend Fáry’s theorem to nonplanar graphs. More specifically, we study the problem of drawing 1plane graphs with straightline edges. A 1plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1plane graphs that admit a straightline drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1plane graphs for which every straightline drawing has exponential area. To our best knowledge, this is the first result to extend Fáry’s theorem to nonplanar graphs. 1
Algorithms For Graphs Embeddable With Few Crossings Per Edge
 PROC. 15TH INT. SYMP. ON FUNDAMENTALS OF COMPUTATION THEORY (FCT’05), VOLUME 3623 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings. We prove that many optimization problems, including maximum independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximati ..."
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Cited by 13 (0 self)
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We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings. We prove that many optimization problems, including maximum independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximation schemes when restricted to such graphs. This extends previous results by Baker [1] and Eppstein [7] to a much broader class of graphs. We also show that testing if a graph can be drawn in the plane with at most one crossing per edge is NPcomplete.
New Bounds on Crossing Numbers
, 1999
"... The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by #(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of P. Erdos and R. Guy by showing that #(n, e)n 2 /e 3 tends ..."
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Cited by 12 (4 self)
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The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by #(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of P. Erdos and R. Guy by showing that #(n, e)n 2 /e 3 tends to a positive constant as n ## and n # e # n 2 . Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e # 4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce 4 /n 3 (resp. ce 5 /n 4 ), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of M. Simonovits. 1 Introduction Let G be a simple undirected graph with n(G) nodes (vertices) and e(G) edges. A drawing of G in the plane is a m...
CHARACTERISATIONS AND EXAMPLES OF GRAPH CLASSES WITH BOUNDED EXPANSION
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of t ..."
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Cited by 10 (3 self)
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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several lineartime algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of socalled topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded nonrepetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologicallyclosed class, while graphs with bounded crossing number are contained in a minorclosed class.
Halving Point Sets
, 1998
"... Given n points in R d , a hyperplane is called halving if it has at most bn=2c points on either side. How many partitions of a point set (into the points on one side, on the hyperplane, and on the other side) by halving hyperplanes can be realized by an npoint set in R d? ..."
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Cited by 5 (0 self)
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Given n points in R d , a hyperplane is called halving if it has at most bn=2c points on either side. How many partitions of a point set (into the points on one side, on the hyperplane, and on the other side) by halving hyperplanes can be realized by an npoint set in R d?