Results 1  10
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19
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 35 (9 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
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...
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 12 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
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 11 (2 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.
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 4 (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?
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 4 (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
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 4 (2 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.
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 3 (1 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
Testing Maximal 1planarity of Graphs with a Rotation System in Linear Time
"... A 1planar graph is a graph that can be embedded in the plane with at most one crossing per edge. A 1planar graph on n vertices can have at most 4n − 8 edges. It is known that testing 1planarity of a graph is NPcomplete. A 1planar embedding of a graph G is maximal, if no edge can be added witho ..."
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Cited by 2 (0 self)
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A 1planar graph is a graph that can be embedded in the plane with at most one crossing per edge. A 1planar graph on n vertices can have at most 4n − 8 edges. It is known that testing 1planarity of a graph is NPcomplete. A 1planar embedding of a graph G is maximal, if no edge can be added without violating the 1planarity of G. In this paper, we study combinatorial properties of maximal 1planar embeddings. In particular, we show that in a maximal 1planar embedding, the graph induced by the noncrossing edges is spanning and biconnected. Using the properties, we show that the problem of testing maximal 1planarity of a graph G can be solved in linear time, if a rotation system Φ (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1planar embedding ξ of G that preserves the given rotation system Φ, and our algorithm also produces such an embedding in linear time, if it exists. In addition, we establish new bounds on the minimum number of edges of maximal 1plane graphs, showing that for graphs on n vertices this number is between
Iterated PointLine Configurations Grow DoublyExponentially
, 2008
"... Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoičić (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on ..."
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Cited by 1 (0 self)
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Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoičić (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the SzemerédiTrotter Theorem and an analysis of the “minimum degree ” of the growing configuration. Consider the iterative process of constructing points and lines in the real plane given by the following: begin with a set of points P1 = {p1, p2, p3, p4} in the real plane in general position. For each pair of points, construct the line passing through the pair. This will create a set of lines L1 = {ℓ1, ℓ2, ℓ3, ℓ4, ℓ5, ℓ6}. Some of these constructed lines will intersect at points in the plane that do not belong to the set P1. Add any such point to the set P1 to get a new set P2. Now, note that there exist some pairs of points in P2 that do not lie on a line in L1, namely some elements of P2 \ P1. Add these missing lines to the set L1 to get a new set L2. Iterate in this manner, adding points to Pk followed by adding lines to Lk. We assume that the original configuration is such that for every k ∈ N no two lines in Lk are parallel. Now we introduce some notation for this iterative process. The kth stage is defined to consist of these two ordered steps: 1. Add each intersection of pairs of elements of Lk to Pk+1, and 2. Add a line through each of pair of elements of Pk to Lk+1. 1 Under this definition, we say that stage 1 begins with the configuration of four points with six lines and stage k begins with nk points with mk lines. We will denote the set of points at the beginning of stage k by Pk and likewise the set of lines at the beginning of stage k by Lk. There are some trivial bounds on the number of points and lines at stage k that can be obtained with this notation. Since a point in Pk must lie at the intersection of at least two lines of Lk−1 we know that at stage k, there are at most () mk−1 points. 2 Similarly, since a line in Lk must contain at least two points from Pk we know that at stage k there are at most ( nk 2 mk−1 nk ≤ From this it follows that 2 lines. In other words,