M.R. Garey and D. S. Johnson. "Crossing number is NP-complete". SIAM Journal on Algebraic and Discrete Methods, 4:312-316, 1983. Home/Search Document Not in Database Summary Related Articles Check |
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On Eggleton And Guy's Conjectured Upper Bound For The.. - Faria, de Figueiredo (2000) (1 citation) (Correct)
....to be simple. A drawing of a graph G is optimum when it has the minimum number of crossings among all drawings of G . This number is called the crossing number of G and is denoted by (G) The algorithmic problem of computing the crossing number of a graph has been shown to be NP complete ([5]) Let Q n denote the n dimensional cube. The vertices of Q n are all n tuples of 0 s and 1 s, of which there are jV (Q n )j = 2 . Two vertices x = x 1 ; x ; x n ) 1991 Ma t h e ma t i c s Sub j e c t C l a s s i f i c a t i o n: Primary 05C10. K e y wo r d s: topological graph ....
GAREY, M. R.---JOHNSON, D. S. : Crossing number is NP-complete, SIAM J. Discrete Math. 4 (1983), 312--316.
Which Crossing Number is It, Anyway? - Courant (Correct)
....with this de nition, because it can be interpreted in several ways. Sometimes it is assumed that in a proper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common ( WB78] B91] Many authors do not make this assumption ( T70] [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 Supported by NSF grant CCR 94 24398 and PSC CUNY Research Award 667339. Supported ....
....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 bipartite graph K n;n with 2n vertices, as n tends to in nity [RT97] The latter question, raised more than fty years ago, is often referred ....
[Article contains additional citation context not shown here]
M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (1983), 312-316.
The Splitting Number and Skewness of - Theta Cm Candido (Correct)
....Previous results The problems of verifying and computing the invariants sk and sp for general graphs have been shown to be respectively NP complete [11, 14] and MAX SNP hard [6, 10] even for cubic graphs. However, it can be checked in polynomial time whether the skewness sk is equal to a fixed k [13]. We have shown [6] that the same holds for the splitting number sp, by the results of Robertson and Seymour [26] The difficulty in computing the invariants sk and sp for general graphs justifies their analysis for special families of graphs. Exact explicit formulas have been found for the ....
M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic and Discrete Methods, 1:312--316, 1983.
On a Matching Problem in the Plane - Dumitrescu, Steiger (2000) (2 citations) (Correct)
....for the case of two colors. Decide if there exists n 0 2 N such that for every even n n 0 ; f(n) g(n) If true, clearly n 0 4. What is the complexity of computing F (S) and f(S) If this is hard as we believe (many problems dealing with crossing properties are NP hard, see e.g. [4], 3] 5] 7] approximating f(S) would be another problem to consider. Acknowledgments. We would like to thank Michael Saks and Leonid Khachiyan for their valuable ideas and comments. 11 ....
M. R. Garey, D. S. Johnson, Crossing number is NP-complete, SIAM Journal on Algebraic and Discrete Methods 4 (1983), 312-316.
Two-Layer Planarization in Graph Drawing - Mutzel, Weiskircher (1998) (4 citations) (Correct)
....In practice, this is done layerwise. Keep the permutation of one layer fix while permuting the other one, such that the number of crossings is reduced. We suggest an alternative approach for the second step. Already for two layer graphs the straight line crossing minimization problem is NP hard [6] even if one layer is fixed [5] Exact algorithms based on branchand bound have been suggested byvarious authors (see, e.g. 9] For k 2, a vast amount of heuristics has been published in the literature (see, e.g. 14] and [3] A new approachisto remove a minimal set of edges such that the ....
M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4:312--316, 1983.
Which Crossing Number is It Anyway? - Pach, Toth (Correct)
....with this definition, because it can be interpreted in several ways. Sometimes it is assumed that in a proper drawing no two edges cross more than once, and if two edges share an endpoint, they cannot have another point in common ( WB78] B91] Many authors do not make this assumption ( T70] [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 # Supported by NSF grant CCR 94 24398 and PSC CUNY Research Award 667339. Supported ....
....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 bipartite graph K n,n with 2n vertices, as n tends to infinity [RT97] The latter question, raised more than fifty years ago, is often referred ....
[Article contains additional citation context not shown here]
M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (1983), 312--316.
On Conway's Thrackle Conjecture - Lovász, Pach, Szegedy (1997) (5 citations) (Correct)
....we do not how to nd the crossing number of G, i.e. the minimum number of crossing pairs of edges in a planar drawing of G. In the case when G is a complete bipartite graph, this is Tur an s brick factory problem [T77, G72] The determination of the crossing number is known to be NP complete [GJ83]. Another well known open problem that illustrates our ignorance about graph drawings was raised by Conway about forty years ago. He de ned a thrackle as a drawing of a graph G with the property that any two distinct edges either (i) share an endpoint, and then they do not have any other point ....
M.R. Garey and D.S. Johnson, Crossing number is NP{complete, SIAM J. Algebraic Discrete Methods 4, 312-316.
Inserting an Edge Into a Planar Graph - Gutwenger, Mutzel, Weiskircher (2000) (1 citation) (Correct)
....which is able to find a crossing minimum solution. 1 Introduction Crossing minimization is among the most challenging problems in graph theory and graph drawing. Although, there is a vast amount of literature on this NP hard problem (for a survey see, e.g. 13] NPhardness is shown in [5]) so far no practically efficient exact algorithm for crossing minimization is known. Currently, the best known approach for crossing minimization is based on planarization. Here, in a first step, the minimum number of edges is deleted so that the resulting graph is planar. Then, the edges are ....
M. R. Garey and D. S. Johnson. Crossing num- ber is NP-complete. SIAM Journal Alg. Disc. Methods, 4:312-316, 1983.
New Bounds on Crossing Numbers - Pach, Tóth (1999) (5 citations) (Correct)
....is also called an edge, and if this leads to no confusion, it is also denoted by uv. We assume that no three edges have an interior point in common. The crossing number, cr(G) of G is the minimum number of crossing points in any drawing of G. The determination of cr(G) is an NP complete problem [GJ83]. It was discovered by Leighton [L84] that the crossing number can be used to estimate the chip area required for the VLSI circuit Supported by NSF grant CCR 97 32101 and PSC CUNY Research Award 667339. Supported by DIMACS Center, OTKA T 020914, and OTKA F 22234. layout of a graph. He proved ....
M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (1983), 312-316.
Mathematical Programming Formulation of Rectilinear Crossing.. - Dean (2002) (Correct)
....(rectilinear) drawing of G with as few edge crossings as possible. This minimum value is called the (rectilinear) crossing number cr(G) respectively, cr(G) of G. Both of the corresponding decision problems are known to be NP hard. Moreover, the crossing number problem is known to be NP complete [16], but so far no one has determined whether the rectilinear crossing number problem is in NP. This is somewhat surprising, since it is much easier to determine if two straight line segments intersect than it is to determine if two curved lines do. This paper presents some new results on ....
M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods, 4 (1983) 312-316.
Crossing Minimization for Symmetries - Buchheim, Hong (2002) (Correct)
....if both problems are restricted to re ections or rotations of xed order. Proof. See App. A. Theorem 1. The problems (SCM) and (SCM ) are NP hard, even if restricted to re ections or rotations of xed order. Proof. We can easily reduce the NP hard problem of crossing minimization for graphs [7] to the crossing minimization problem for re ections or rotations of xed order k. For this, let G be any graph. Construct a new graph G as the disjoint sum of k copies of G. De ne by mapping the copies cyclically to each other. Obviously, drawing with a minimal number of edge crossings is ....
M. Garey and D. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods, 4:312-316, 1983.
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M.R. Garey and D. S. Johnson. "Crossing number is NP-complete". SIAM Journal on Algebraic and Discrete Methods, 4:312-316, 1983.
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M. R. Gary and D. S. Johnson. (1983), Crossing number is NP-complete, SIAM J. Algeraic and Discrete Methods, 4, 312--316.
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M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods, 4:312-316, 1983.
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M. Garey and D. Johnson, "Crossing Number is NP-Complete," SIAM Journal of Algebraic and Discrete Methods, vol. 4, no. 3, pp. 312-316, 1983.
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M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods, 4:312-316, 1983.
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M. R. Garey and D. S. Johnson, Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4(3):312-316, 1983.
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M.R. Garey, D.S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic and Discrete Methods 4:312-316, (1983).
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M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983), 312--316.
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Michael R. Garey and David S. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods, 4(3):312--316, 1983.
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M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods, 4(3):312--316, 1983.
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