D. Bienstock, N. Dean, Bounds for rectilinear crossing numbers, J.Graph Thph 17(3) (1993) 333--348.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on the drawings (e.g. the edges Supported by NSF grant CCR 94 24398 and PSC CUNY Research Award 667339. Supported by OTKA T 020914, OTKA F 22234, and the Margaret and Herman Sokol Posdoctoral Fellowship Award. must be straight line segments [J71] or polygonal paths of length at most k [BD93]) No matter what de nition we use, the determination of the crossing number of a graph appears to be an extremely dicult task ( GJ83] B91] In fact, we do not even know the asymptotic value of any of the above quantities for the complete graph K n with n vertices and for the complete ....

....now crossings are counted without multiplicities. iv) The odd crossing number of G, odd cr(G) is the minimum number of pairs of edges (e; e such that e and e cross an odd number of times. Clearly, we have odd cr(G) pair cr(G) cr(G) lin cr(G) It was shown by Bienstock and Dean [BD93] that there are graphs with crossing number 4, whose rectilinear crossing numbers are arbitrarily large. On the other hand, we cannot rule out the possibility that odd cr(G) pair cr(G) cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani ....

D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993), 333-348. 19

....on the drawings (e.g. the edges # Supported by NSF grant CCR 94 24398 and PSC CUNY Research Award 667339. Supported by OTKA T 020914, OTKA F 22234, and the Margaret and Herman Sokol Posdoctoral Fellowship Award. must be straight line segments [J71] or polygonal paths of length at most k [BD93]) No matter what definition we use, the determination of the crossing number of a graph appears to be an extremely di#cult task ( GJ83] B91] In fact, we do not even know the asymptotic value of any of the above quantities for the complete graph K n with n vertices and for the complete ....

....of G. That is, now crossings are counted without multiplicities. iv) The odd crossing number of G, odd cr(G) is the minimum number of pairs of edges (e, e # ) such that e and e # cross an odd number of times. Clearly, we have pair cr(G) lin cr(G) It was shown by Bienstock and Dean [BD93] that there are graphs with crossing number 4, whose rectilinear crossing numbers are arbitrarily large. On the other hand, we cannot rule out the possibility that odd cr(G) pair cr(G) cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani ....

D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993), 333--348. 19

....drawing. In App. B, we show that every planar symmetry admits a planar straight line drawing. Hence, planarity of symmetries does not depend on whether we require straight lines or not. This is not true for the crossing number of a symmetry, which follows from the corresponding result for graphs [2] but is obvious even for symmetries with a single orbit; see Fig. 5. Fig. 5. Intra orbit edges may be drawn inside or outside their orbit circle. In general, requiring all edges to be drawn on the same side, for example in straight line drawings, increases the number of necessary edge crossings ....

D. Bienstock and N. Dean. Bounds for rectilinear crossing numbers. Journal of Graph Theory, 17:333-348, 1993.

....of determining the crossing number of an arbitrary graph is NP complete. Leighton [Lei84] gave an application to VLSI design by demonstrating a relationship between the area required to design a chip whose circuit is given by the graph G and the rectilinear crossing number of G. Bienstock and Dean [BD93] produced an infinite family of graphs Gm with #(Gm ) 4 for every m but for which supm #(Gm ) #. Kleitman [Kle70, Kle76] completed the very di#cult task of determining the exact value of #(K 5,n ) for any n # # . Finally, a crucial method of attack for both rectilinear crossing ....

D. Bienstock and N. Dean. Bounds for rectilinear crossing numbers. Journal of Graph Theory, 17(3):333--348, 1993.

....of determining the crossing number of an arbitrary graph is NP complete. Leighton [Lei84] gave an application to VLSI design by demonstrating a relationship between the area required to design a chip whose circuit is given by the graph G and the rectilinear crossing number of G. Bienstock and Dean [BD93] produced an in nite family of graphs fGm g with (Gm ) 4 for every m but for which supm f (Gm )g = 1: Kleitman [Kle70, Kle76] completed the very dicult task of determining the exact value of (K 5;n ) for any n 2 Z . Finally, a crucial method of attack for both rectilinear crossing number ....

D. Bienstock and N. Dean. Bounds for rectilinear crossing numbers. Journal of Graph Theory, 17(3):333-348, 1993.

....that cr(K n ) cr(K n ) for n 10 but there does not seem to be a reasonable conjecture on the values cr(K n ) for n 10. Finally, it can be shown that lim n 1 cr(Kn ) n 4 and lim n 1 cr(Kn ) n 4 exist (see [25] and it is conjectured that both limits are equal to 1 64 . Following [7], there exist graphs with crossing number greater than or equal to four and arbitrarily high rectilinear crossing number and, on the other hand, for every graph G with crossing number cr(G) 3, the equality cr(G) cr(G) holds. Graphs on Surfaces By well known theorems of Kuratowski and Wagner, ....

D. Bienstock and N. Dean, Bounds for Rectilinear Crossing Numbers, J. Graph Theory 17 (1993), 333--348.

....of G is drawn as a straight line segment. Clearly #(G) # #(G) For graphs with bounded degree, the crossing number and the rectilinear crossing number are bounded functions of one another [BD92] SSSV95, SSSV96a] But for every m k # 4, there exists a graph G with #(G) k and #(G) # m [BD93] So the rectilinear crossing number can be arbitrarily large in comparison to the crossing number. A further restriction of the rectilinear crossing number yields the following problem: The vertices of the graph under consideration are partitioned into k # 2 classes (usually called layers in ....

Daniel Bienstock and Nathaniel Dean. Bounds for rectilinear crossing numbers. J. of Graph Theory, 17:333--348, 1993.

....is true for complete graphs. For the crossing number [11, 18] which like the thickness is a measure of how far a graph is from being planar, the analogous question is known to have a negative answer. M. B. Dillencourt et al. Geometric Thickness , JGAA, 4(3) 5 17 (2000) 16 Bienstock and Dean [7] have described families of graphs which have crossing number 4 but arbitrarily high rectilinear crossing number (where the rectilinear crossing number is the crossing number restricted to drawings in which all edges are line segments) 4. What is the complexity of computing #(G) for a given ....

D. Bienstock and N. Dean. Bounds for rectilinear crossing numbers. Journal of Graph Theory, 17(3):333--348, 1993.

....of determining the crossing number of an arbitrary graph is NP complete. Leighton [Lei84] gave an application to VLSI design by demonstrating a relationship between the area required to design a chip whose circuit is given by the graph G and the rectilinear crossing number of G. Bienstock and Dean [BD93] produced an infinite family of graphs fGm g with cr(Gm ) 4 for every m but for which supm fcr(Gm )g = 1: Kleitman [Kle70, Kle76] completed the very difficult task of determining the exact value of cr(K 5;n ) for any n 2 Z . Finally, a crucial method of attack for both rectilinear crossing ....

D. Bienstock and N. Dean. Bounds for rectilinear crossing numbers. Journal of Graph Theory, 17(3):333--348, 1993.

....precision edge crossing counter. 10 [Gar86, Sin71] was the first successful recorded instance of this break with tradition. Additionally, can the technique given in Section 3 be applied successfully to other families of interesting graphs See, for example, the work of Bienstock and Dean [BD93, BD92]. Our second open question: given the current status of computing it may be possible to apply brute force techniques to determine the exact value of cr(K n ) for small values of n beyond what is presently known [Guy72, WB78] Design and implement an algorithm that produces all (non isomorphic) ....

D. Bienstock and N. Dean. Bounds for rectilinear crossing numbers. Journal of Graph Theory, 17(3):333--348, 1993.

....that each edge of G is drawn as a straight line segment. Clearly (G) G) For graphs with bounded degree, the crossing number and the rectilinear crossing number are bounded functions of one another [BD92, SSSV95] But for every m k 4, there exists a graph G with (G) k and (G) m [BD93] So the rectilinear crossing number can be arbitrarily large in comparison to the crossing number. A further restriction of the rectilinear crossing number yields the following problem: The vertices of the graph under consideration are partitioned into k 2 classes (usually called layers in ....

Daniel Bienstock and Nathaniel Dean. Bounds for rectilinear crossing numbers. J. of Graph Theory, 17:333-348, 1993.

....[GJ83] SSSV97] If two edges are allowed to cross several times, we may count their intersections with multiplicity or without. We may also wish to impose some further restrictions on the drawings (e.g. the edges must be straight line segments [J71] or polygonal paths of length at most k [BD93]) No matter what definition we use, the determination of the crossing number of a graph appears to be an extremely difficult task ( GJ83] B91] In fact, we do not even know the asymptotic value of any of the above quantities for the complete graph K n with n vertices and for the complete ....

....are counted without multiplicities. iv) The odd crossing number of G, odd cr(G) is the minimum number of pairs of edges (e; e 0 ) such that e and e 0 cross an odd number of times. Clearly, we have odd cr(G) pair cr(G) cr(G) lin cr(G) 2 It was shown by Bienstock and Dean [BD93] that there are graphs with crossing number 4, whose rectilinear crossing numbers are arbitrarily large. On the other hand, we cannot rule out the possibility that odd cr(G) pair cr(G) cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani ....

D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993), 333--348.

....number. For instance the planar crossing number of K 8 is 18, but its rectilinear crossing number is 19 [9] The gap between the planar crossing number and the rectilinear crossing number can not be bounded by a function of only the planar crossing number, as demonstrated by Bienstock and Dean [7]. Thus the value of the planar crossing number does not provide for a good approximation to the rectilinear crossing number. In particular, the curves used for drawing edges of G employing Theorem 4.1 may have many bends, and thus drawings obtained through the method of Theorem 4.1 are not ....

....number does not provide for a good approximation to the rectilinear crossing number. In particular, the curves used for drawing edges of G employing Theorem 4.1 may have many bends, and thus drawings obtained through the method of Theorem 4.1 are not straight lines drawings. Bienstock and Dean [6, 7] have extensively investigated the relationship between cr 0 (G) and the rectilinear crossing number of G, cr 0 (G) and have obtained the following. Theorem 4.2 [6] For any graph G, we have, cr 0 (G) O( Deltacr 0 (G) 2 ) 2 The algorithm used to construct the desirable drawing in Theorem 4.2 ....

D. Bienstock and N. Dean, Bounds on the rectilinear crossing numbers, J. Graph Theory 17 (1993) 333-348.

....pointing out some di#erences between these two parameters and some open problems on the rectilinear crossing number. Along the way we examine some other more specialized crossing number problems. First, here s a question we can answer. Is the di#erence cr(G) cr(G) unbounded Bienstock and Dean [9] showed that, for every integer k 4, there is an infinite family of graphs of crossing number k, but unbounded rectilinear crossing number. On the other hand, they prove that any graph G for which cr(G) 3 must also satisfy cr(G) cr(G) an extension of the classic theorem of Steinitz and ....

D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory, 17 (1993), no. 3, 333--348.

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D. Bienstock, N. Dean, Bounds for rectilinear crossing numbers, J.Graph Thph 17(3) (1993) 333--348.

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D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17(1993), 333--348. 3

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