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

2003
*
Discrete Mathematics
*

For each n we construct a

doi:10.1016/s0012-365x(02)00491-0
fatcat:kmk5yc2ifbazjcg3mfcgpila5a
*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] . ...##
###
The 2-page crossing number of Kn

2012
*
Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12
*

Around 1958, Hill conjectured that

doi:10.1145/2261250.2261310
dblp:conf/compgeom/AbregoAFRS12
fatcat:avzuef2lhbdwpcxbcmnck6wqdu
*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*...##
###
Improved Bounds for the Crossing Numbers of Km,n and Kn

2006
*
SIAM Journal on Discrete Mathematics
*

In this paper we show

doi:10.1137/s0895480104442741
fatcat:oyjawueulzdvnii4mmpqjzd7uu
*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 ...##
###
New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn

2007
*
Discrete & Computational Geometry
*

We extend

doi:10.1007/s00454-007-1325-8
fatcat:4nliyppvkvb2vowq73x4ixmwjm
*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. ...##
###
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]

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] . ...

##
###
Maximizing the Total Resolution of Graphs

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] . ...

##
###
Geometric Data Structures
[chapter]

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 ...

##
###
Planarizing Graphs - A Survey and Annotated Bibliography

2001
*
Journal of Graph Algorithms and Applications
*

While there are many algorithmic results about planarization through edge deletion,

doi:10.7155/jgaa.00032
fatcat:zftlx3a5jnesxff5hjymjewp5u
*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] . ...##
###
Straight skeletons of three-dimensional polyhedra

2009
*
Proceedings of the 25th annual symposium on Computational geometry - SCG '09
*

We analyze

doi:10.1145/1542362.1542384
dblp:conf/compgeom/BarequetV09
fatcat:g3jl2gm4anh3tdmm2ahbzknawm
*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.) ...##
###
Computing the Largest Bond and the Maximum Connected Cut of a Graph
[article]

2020
*
arXiv
*
pre-print

Finally, we show that both problems are fixed-parameter tractable when parameterized by

arXiv:2007.04513v1
fatcat:c2rnzrss2jdehmuog7hgmdjmzy
*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. ...##
###
Straight Skeletons of Three-Dimensional Polyhedra
[article]

2008
*
arXiv
*
pre-print

We analyze

arXiv:0805.0022v1
fatcat:rmhhdcsv6ncsjfo3pawovzkcle
*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.) ...##
###
THE HAUSDORFF VORONOI DIAGRAM OF POLYGONAL OBJECTS: A DIVIDE AND CONQUER APPROACH

2004
*
International journal of computational geometry and applications
*

Specifically, we show that for arbitrary polygons or arbitrary clusters

doi:10.1142/s0218195904001536
fatcat:jitnfhrk5ngupnuyl7bt7oxpsi
*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). ...##
###
The Graph Crossing Number and its Variants: A Survey

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 ...

##
###
The Graph Crossing Number and its Variants: A Survey

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. ...

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