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Toward the rectilinear crossing number of Kn: new drawings, upper bounds, and asymptotics

Alex Brodsky, Stephane Durocher, Ellen Gethner
2003 Discrete Mathematics  
For each n we construct a rectilinear drawing of K_n that has the fewest number of edge crossings and the best asymptotics known to date.  ...  The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n.  ...  Acknowledgements The authors wish to thank David Kirkpatrick and Nick Pippenger for stimulating discussions and ideas. We are also grateful to David Singer who provided us with his manuscript [21] .  ... 
doi:10.1016/s0012-365x(02)00491-0 fatcat:kmk5yc2ifbazjcg3mfcgpila5a

The 2-page crossing number of Kn

Bernardo Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar
2012 Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12  
Around 1958, Hill conjectured that the crossing number cr(K n  ...  This inequality used with the rectilinear version of Theorem 1 gives Z (n) as a lower bound for the rectilinear crossing number of K n [1] .  ...  We recall that a drawing D is rectilinear if the edges of D are straight line segments, and the rectilinear crossing number cr(G) of a graph G is the minimum of cr(D) taken over all rectilinear drawings  ... 
doi:10.1145/2261250.2261310 dblp:conf/compgeom/AbregoAFRS12 fatcat:avzuef2lhbdwpcxbcmnck6wqdu

Improved Bounds for the Crossing Numbers of Km,n and Kn

E. de Klerk, J. Maharry, D. V. Pasechnik, R. B. Richter, G. Salazar
2006 SIAM Journal on Discrete Mathematics  
In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_n->infty cr(K_m,n)/Z(m,n) >= 0.83m/(m  ...  The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_7,n) >= 2.1796n^2 - 4.5n.  ...  In section 4 we discuss consequences of Theorem 1: The improved bound for cr(K 7,n ) implies improved asymptotic bounds for the crossing numbers of cr(K m,n ) and cr(K n ). 1 3 4 5 2 6) and (0 2 6  ... 
doi:10.1137/s0895480104442741 fatcat:oyjawueulzdvnii4mmpqjzd7uu

New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn

Oswin Aichholzer, Jesus Garcia, David Orden, Pedro Ramos
2007 Discrete & Computational Geometry  
We extend the range of known values for the rectilinear crossing number, namely by cr(K 19 ) = 1318 and cr(K 21 ) = 2055.  ...  We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position.  ...  Acknowledgements The authors would like to thank the anonymous referees for several comments that helped improving this paper.  ... 
doi:10.1007/s00454-007-1325-8 fatcat:4nliyppvkvb2vowq73x4ixmwjm

Page 6543 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
(3-BC-C; Vancouver, BC); Durocher, Stephane (3-BC-C; Vancouver, BC); Gethner, Ellen (3-BC-C; Vancouver, BC) Toward the rectilinear crossing number of K,,: new drawings, upper bounds, and asymptotics.  ...  This beautiful and exciting paper improves on the best known asymptotic bounds of the rectilinear crossing numbers of the com- plete graphs.  ... 

Maximizing the Total Resolution of Graphs [article]

Evmorfia N. Argyriou, Michael A. Bekos, Antonios Symvonis
2010 arXiv   pre-print
The main contribution of the paper consists of drawings of asymptotically optimal total resolution for complete graphs (circular drawings) and for complete bipartite graphs (2-layered drawings).  ...  In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution.  ...  For this class of graphs, they proved upper and lower bounds for the number of edges. Force-directed methods are commonly used for drawing graphs [8, 10] .  ... 
arXiv:1009.2109v2 fatcat:c3ag3nh2unbfdmqo2rfsjtolue

Maximizing the Total Resolution of Graphs

E. N. Argyriou, M. A. Bekos, A. Symvonis
2012 Computer journal  
The main contribution of the paper consists of drawings of asymptotically optimal total resolution for complete graphs (circular drawings) and for complete bipartite graphs (2-layered drawings).  ...  In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution.  ...  For this class of graphs, they proved upper and lower bounds for the number of edges. Force-directed methods are commonly used for drawing graphs [8, 10] .  ... 
doi:10.1093/comjnl/bxs088 fatcat:oj6v7w3k25aspcuioxziexrske

Geometric Data Structures [chapter]

Michael T. Goodrich, Kumar Ramaiyer
2000 Handbook of Computational Geometry  
The arcs from the boxes to the sink also have (besides the upper bound of 2n/r on the flow) a lower bound of n/2r on the flow.  ...  Thus from a worst-case and asymptotic point of view the problem of computing rectilinear r-partitions with low stabbing number is solved.  ...  We obtain the following two results: (i) an O(m) time algorithm that, given F and a maximum error , computes a function F with the minimum number of links such that error(F, F) ; (ii) an O(n 4/3+δ + m  ... 
doi:10.1016/b978-044482537-7/50011-5 fatcat:ci43rzsdxjeodormvg4q7oujz4

Planarizing Graphs - A Survey and Annotated Bibliography

Annegret Liebers
2001 Journal of Graph Algorithms and Applications  
While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature.  ...  We give a survey of results about such operations and related graph parameters.  ...  For graphs with bounded degree, the crossing number and the rectilinear crossing number are bounded functions of one another [BD92] [SSSV95, SSSV96a] .  ... 
doi:10.7155/jgaa.00032 fatcat:zftlx3a5jnesxff5hjymjewp5u

Straight skeletons of three-dimensional polyhedra

Gill Barequet, Amir Vaxman
2009 Proceedings of the 25th annual symposium on Computational geometry - SCG '09  
We analyze the ways in which the skeleton may intersect each voxel of the polyhedron, and show that the skeleton may be constructed by a simple voxel-sweeping algorithm taking constant time per voxel.  ...  We also consider more general polyhedra with axisparallel edges and faces, and show that any n-vertex polyhedron of this type has a straight skeleton with O(n 2 ) features.  ...  Figure 8shows the straight skeletons of a few simple objects, and the performance of our implementation. (Note that object (e) contains three hole polygons in addition to the 15 facets.)  ... 
doi:10.1145/1542362.1542384 dblp:conf/compgeom/BarequetV09 fatcat:g3jl2gm4anh3tdmm2ahbzknawm

Computing the Largest Bond and the Maximum Connected Cut of a Graph [article]

Gabriel L. Duarte, Hiroshi Eto, Tesshu Hanaka, Yasuaki Kobayashi, Yusuke Kobayashi, Daniel Lokshtanov, Lehilton L. C. Pedrosa, Rafael C. S. Schouery, Uéverton S. Souza
2020 arXiv   pre-print
Finally, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, the treewidth, and the twin-cover number.  ...  Contrasting with a large number of studies related to maximum cuts, there exist very few results regarding the largest bond of general graphs.  ...  Notice that we can draw H φ in the plane according to a monotone rectilinear representation.  ... 
arXiv:2007.04513v1 fatcat:c2rnzrss2jdehmuog7hgmdjmzy

Straight Skeletons of Three-Dimensional Polyhedra [article]

Gill Barequet, David Eppstein, Michael T. Goodrich, Amir Vaxman
2008 arXiv   pre-print
We analyze the ways in which the skeleton may intersect each voxel of the polyhedron, and show that the skeleton may be constructed by a simple voxel-sweeping algorithm taking constant time per voxel.  ...  We also consider more general polyhedra with axis-parallel edges and faces, and show that any n-vertex polyhedron of this type has a straight skeleton with O(n^2) features.  ...  Figure 8 shows the straight skeletons of a few simple objects, and the performance of our implementation. (Note that object (e) contains one hole polygon in addition to the 9 facets.)  ... 
arXiv:0805.0022v1 fatcat:rmhhdcsv6ncsjfo3pawovzkcle

THE HAUSDORFF VORONOI DIAGRAM OF POLYGONAL OBJECTS: A DIVIDE AND CONQUER APPROACH

EVANTHIA PAPADOPOULOU, D. T. LEE
2004 International journal of computational geometry and applications  
Specifically, we show that for arbitrary polygons or arbitrary clusters of points, the size of the Hausdorff Voronoi diagram is O(n+m), where m is O(n 2 ) and reflects the number of crossings among shapes  ...  In this paper we list the structural properties of the Hausdorff Voronoi diagram and provide tighter combinatorial bounds and algorithms.  ...  Thus, the number of non-crossing mixed Voronoi vertices on T (P ) is upper-bounded by |T (P )|. Hence, the total number of non-crossing mixed Voronoi vertices is O(n).  ... 
doi:10.1142/s0218195904001536 fatcat:jitnfhrk5ngupnuyl7bt7oxpsi

The Graph Crossing Number and its Variants: A Survey

Marcus Schaefer
unpublished
The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number  ...  We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing  ...  Acknowledgments Thanks to Markus Chimani, Jan Kynčl, Gelasio Salazar and Paul Kainen for corrections and comments, Heiko Harborth for supplying me with many sources which I had missed, and the anonymous  ... 
fatcat:awqboxa3j5do3d43utk64ixuxa

The Graph Crossing Number and its Variants: A Survey

Marcus Schaefer
2013 unpublished
The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number  ...  We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing  ...  Acknowledgments Thanks to Markus Chimani for several corrections, and the anonymous referees for many helpful suggestions.  ... 
fatcat:it6co5vw25dg7dyuacqbo4ui54
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