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

....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.

....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 ....

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M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (1983), 312-316.

....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.

....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.

....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.

....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.

....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.

....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.

....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.

....(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.

....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.

.... nal layout is usually split into several sub tasks (for an overview of graph drawing see [1] One very important step in the overall ow of graph drawing is crossing minimization, see e.g. 5, 20, 21] Unfortunately, even minimizing edge crossings in graphs with only two layers is NP hard [7] and remains NP hard even if the positions of the nodes in one layer are xed. Several heuristic methods have been developed [4, 8, 12, 15, 18, 20] but exact solutions can be found only for small graphs [10, 11] Almost all heuristic methods are based on the so called layer by layer sweep: ....

M. R. Garey, and D.S. Johnson. Crossing number is NP-Complete. SIAM Journal on Algebraic and Discrete Methods, 4:312-316,1983.

....on the available space is in fact entailed by the second and third criteria stated above. In general, the optimization problems associated with most esthetics are NPhard [10, 11] For instance, it has been proven that even just minimizing the number of edge crossings is an NP hard problem [6]. Therefore the problem described above is also NP hard, thus providing a valid motivation for resorting to an evolutionary approach. The use of evolutionary algorithms for drawing directed graphs, a problem of great practical importance, already began to be explored by Michalewicz [12] ....

M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods, 4(3):312--316, 1983.

....University of Sydney, Australia. the number of layers it crosses; the total span of the edges; and the maximum number of vertices in one layer. Unfortunately, the question of whether a graph G can be drawn in two layers with at most k crossings, where k is part of the input, is NP complete [EW94b,GJ83] as is the question of whether r or fewer edges can be removed from G so that the remaining graph has a crossing free drawing on two layers [TKY77,EW94a] Both problems remain NP complete when the permutation of vertices in one of the layers is given [EW94b,EW94a] When, say, the maximum number ....

M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4(3):312-316, 1983.

....on this problem as well. Instead of deleting edges, one can seek to minimize the number of crossings in a 2 layer drawing (here the input graph must be bipartite) The corresponding problems are called 1 and 2 Layer Crossing Minimization. Both of these well studied problems are NP complete [7, 6]. The 2 Layer Planarization problem is NP complete [5, 17] even for planar biconnected bipartite graphs with vertices in respective bipartitions having degree two and three [5] The 1 Layer Planarization problem is NP complete even for graphs with only degree 1 vertices in the xed layer and ....

M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4(3):312-316, 1983.

....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 94 24398 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.

....in each layer V i are drawn on a horizontal line L i with y coordinate k Gamma i, and the edges are drawn as straight lines. Essentially, a k layered network is a k partite graph that is drawn in a special way. Even for 2 layered graphs the straightline crossing minimization problem is NP hard [9]. The problem consists of aligning the two shores V 1 and V 2 of a bipartite graph G = V 1 ; V 2 ; E) on two parallel straight lines (layers) such that the number of crossings between the edges in E is minimized when the edges are drawn as straight lines. Let n 1 = jV 1 j, n 2 = jV 2 j, m = jEj, ....

M. Garey and D. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4:312--316, 1983.

....9. A layout for a triangular grid graph. 11 We end this chapter with some remarks concerning the Edge Crossing Problem (ECP) Given an undirected graph G, ECP is the problem of determining the minimum number of edge crossings (denoted by (G) among layouts of G. ECP is known to be NP complete [8]. The following approximation is known for the crossing number of complete bipartite graphs [t0, p. 123] v(Km,n) t] TimGA easily reaches the above bound for graphs Km,m, where m 12. Figure t0 shows a drawing for K8,8. Figure 10. A layout for K8,8. 7. Conclusions TimGA nicely draws ....

M. R. Garey and D. S. Johnson, Crossing number is NP-complete. SIAM J. Alg. Disc. Meth. 4, 3 (September 1983), 312-316.

....not intersect, if such an embedding exists. This problem is known as level planar embedding problem. If such a planar embedding is not possible, then we may want to determine a level embedding with as few crossing arcs as possible. This second crossing minimization problem is known to be NP hard [4], even if there are only two levels with a xed order of nodes on one level as shown by Eades and Wormald [3] while the rst mentioned planarity test can be performed in linear time by an involved algorithmic approach [11] In this note we present formulations of these problems in terms of CNF ....

.... avour are known as Partial MAXSAT , i.e. an instance of Partial MAXSAT consists of two formulas f A and f B , and its feasible solutions must satisfy all clauses of f A and as many clauses of f B as possible. As already mentioned the crossing minimization problem is known to be NP hard [4], even if there are only two levels with a xed order of nodes on one level as shown by Eades and Wormald [3] Therefore, a lot of e ort (see [2] has been spent to develop ecient heuristics for reducing the number of crossings in drawings of 2 level graphs and to perform a layer by layer sweep ....

M. R. Garey, D. S. Johnson, Crossing number is NP-complete, SIAM Journal on Algebraic and Discrete Methods, 4(3) (1983), pp. 312-316.

....is called optimal. 1.1 A Few General Results We mention a small variety of papers on crossing numbers problems for graphs drawn in the plane that merely hint at the proliferation of available (and unavailable ) results. Other important results will be highlighted in Section 6. Garey and Johnson [GJ83] showed that the problem 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 ....

....the optimal drawing unique 5. Let G be an arbitrary graph. What is the complexity of determining #(G) Similarly, where in the complexity hierarchy does the determination of #(K n ) live Recall that for a not necessarily rectilinear drawing of G, the general problem is known to be NP complete [GJ83]. For some thoughts on such problems, see [Bie91] 6. We have seen that #(K 11 ) # 98, 100, 102 and believe #(K 11 ) to be 102. Give a combinatorial proof. 7. Finally, in the spirit of the present paper we feel compelled to mention the following problem, for which we sincerely apologize. ....

M. Garey and D. Johnson. Crossing number is NP-complete. SIAM Journal of Algebraic and Discrete Methods, 4:312--316, 1983.

....drawn as straight lines is minimized. Since the number of crossings depends only on the permutation # i of the nodes on each layer i (and not on their absolute position) we can say that the objective is to find the presentation #G, # 0 ,# 1 # of G that minimizes crossings. Gary and Johnson [5] proved that the two layer edge crossing minimization problem is NP hard. More recently, Eades, McKay, and Wormald [2] proved that the problem is NP hard even if the permutation of nodes on one layer is fixed. The problem of determining the minimum cardinality set of edges whose removal allows G ....

M. R. Garey and D. S. Johnson, Crossing Number is NP-complete, SIAM J. Algebraic Discrete Methods, 4 (1983), pp. 312--316.

....of G in the plane. Crossing number results have been achieved only for a few special families of graphs, e.g. complete and complete bipartite graphs, and even in these cases the given formulas have only been verified for small numbers of vertices [3] For arbitrary G, computing (G) is NP hard [4]. Hence, from a computational standpoint, it is infeasible to obtain exact solutions for graphs, in general, but more practical to explore bounds for the parameter values. Similar bounds have previously been given for two other popular parallel network topologies the hypercube [9, 10] and cube ....

M.R. Garey and D.S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (3) (1983) 312-316.

....an embedding of G is determined in linear time, by applying an embedding algorithm [14, 30] If G is not planar, the minimum number of dummy vertices introduced may be n 4 ) However, in practice this number is usually much smaller. Minimizing the number of crossings is in general NP hard [21]. For a survey on planarization techniques see [17] A popular algorithm for constructing an orthogonal representation of an embedded graph with vertices having at most four incident edges was presented by Tamassia [38] Such an algorithm computes an orthogonal representation that has the minimum ....

M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4(3):312-316, 1983.

.... as much as possible [11] Similar considerations also hold for the design of VLSI circuits [7] Also, automatic graph drawing systems make use of crossing reduction techniques to display graphs which are aesthetically pleasing and more comprehensible [15] The crossing number problem is NP hard [1]; hence research has focused on finding efficient heuristics or on methods for special families of graphs. For arbitrary graphs, one of the first heuristics devised was that of Nicholson [9] An assortment of other methods, invariably based on greedy and local improvement techniques, have also ....

M.R. Garey and D.S. Johnson, "Crossing number is NPcomplete, " SIAM J. Alg. Disc. Meth. 4 (3), pp. 312-316, 1983.

....c Utilitas Mathematica Publishing, Winnipeg, Canada. 1 y u v x y u v x Figure 1: Edge twisting operation. have been determined only for some instances of a few special families of graphs, e.g. complete and complete bipartite graphs [6] For arbitrary G, computing (G) is NP hard [9]. Hence, from a computational standpoint, it is infeasible to obtain exact values for this parameter for graphs, in general, but more practical to nd upper and lower bounds. Such bounds have previously been given for two other popular parallel network topologies the hypercube [8, 15, 18] and ....

Garey, M.R. and D.S. Johnson. Crossing number is NP-complete. SIAM J. Alg. Disc. Meth. 4 (3) (1983) 312-316.

.... There are many practical applications of such drawings, including VLSI design [3] or graph visualization [4, 14] Crossingnumber problems are often discussed on Graph Drawing conferences, recently for example [12, 18, 14] Determining the crossing number of a graph is a hard problem [6] in general, and the crossing number is not even known exactly for complete or complete bipartite graphs. A lot of work has been done investigating the crossing number of particular graph classes like Cm C n , see [15, 16, 8] For general graphs, research so far focused mainly on relations of the ....

M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983), 312-316.

....is called optimal. 1.1 A Few General Results We mention a small variety of papers on crossing numbers problems for graphs drawn in the plane that merely hint at the proliferation of available (and unavailable ) results. Other important results will be highlighted in Section 6. Garey and Johnson [GJ83] showed that the problem 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 ....

....the optimal drawing unique 5. Let G be an arbitrary graph. What is the complexity of determining (G) Similarly, where in the complexity hierarchy does the determination of (K n ) live Recall that for a not necessarily rectilinear drawing of G, the general problem is known to be NP complete [GJ83]. For some thoughts on such problems, see [Bie91] 6. We have seen that (K 11 ) 2 f98; 100; 102g and believe (K 11 ) to be 102. Give a combinatorial proof. 7. Finally, in the spirit of the present paper we feel compelled to mention the following problem, for which we sincerely apologize. Since ....

M. Garey and D. Johnson. Crossing number is NP-complete. SIAM Journal of Algebraic and Discrete Methods, 4:312-316, 1983.

....Note that these strategies do not address the problem of minimizing the number of crossings in the whole graph: even with the restriction of looking at consecutive layers only, minimization of edge crossings is difficult and complex. In fact, Garey and Johnson proved the problem to be NP hard[53] and Eades and Whitesides proved the corresponding decision problem to be NP complete[36] The complexity of a proper minimization has motivated the development of various heuristics for computing a good order for the nodes on a layer. Tutte[119] was the first to propose a heuristics: starting ....

M.R. Garey and D. S. Johnson (1983). "Crossing number is NP-- complete", SIAM Journal of Algebraic and Discrete Methods Vol. 4 No 3, pp. 312--316, 1983.

.... the area of the layout as well as the number of wire contact cuts that should be minimized [1] For this reason deriving asymptotic bounds for the planar crossing number has been popular in the VLSI community [17] Computing the planar crossing number of a graph has been known to be NP complete [8]. A natural generalization of the planar crossing number is the crossing number of a graph on an orientable or nonorientable surface of genus g # 1[13] 14] 23] 28] 1 The research of the third and the fourth authors was partially supported by Grant No. 2 1138 94 of the Slovak Academy of ....

Garey, M. R., and Johnson, D. S., Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods, 4 (1983), 312--316.

....for arbitrarily large crossing numbers are known say White and Beineke [WB78] and we may add that even 15 years later the known ones are for some families of very sparse graphs of strong structure; not even cr 0 (K n ) and cr 0 (K m, n ) are known exactly. Computing cr g (G) is NP hard for g=0 [GJ83]. Therefore, it makes sense to estimate cr g (G) and cr t g (G) under fairly general conditions. We try to list here the most important results on cr g (G) and cr t g (G) that has been known. Ringel [Ri55] and independently Beineke and Harary [BH65] determined the orientable genus of the ....

M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic and Discrete Methods 4 (1983), 312#316.

....9. A layout for a triangular grid graph. 12 We end this chapter with some remarks concerning the Edge Crossing Problem (ECP) Given an undirected graph G, ECP is the problem of determining the minimum number of edge crossings (denoted by n(G) among layouts of G. ECP is known to be NP complete [8]. The following approximation is known for the crossing number of complete bipartite graphs [10, p. 123] v(K m,n ) 2 m 2 m 1 2 n 2 n 1 . TimGA easily reaches the above bound for graphs K m,m , where m 12. Figure 10 shows a drawing for K 8,8 . Figure 10. A layout for K ....

M. R. Garey and D. S. Johnson, Crossing number is NP-complete. SIAM J. Alg. Disc. Meth. 4, 3 (September 1983), 312-316.

.... a Graph We are interested in the following problem: Problem 46 (Crossing Number) Given a graph G and a positive integer K, is there a drawing of G with K or less edge crossings The complexity status of this problem was mentioned as being open in [GJ79, Problem OPEN3] Then Garey and Johnson [GJ83] showed that Crossing Number is NP complete. They use a two step reduction, starting with the following NP complete problem [GJS76] Problem 47 (Optimal Linear Arrangement [GJ79, Problem GT42] Given a graph G = V, E) and a positive integer K, is there a bijection f : V # 1, 2, V ....

.... GT42] Given a graph G = V, E) and a positive integer K, is there a bijection f : V # 1, 2, V such that # uv#E f(u) f(v) # K First, Optimal Linear Arrangement is reduced to a problem introduced as Bipartite Crossing Number: Problem 48 (Bipartite Crossing Number [GJ83] Given a connected bipartite graph G = V 1 , V 2 , E) with multiple edges allowed, and given a positive integer K, can G be drawn in the unit square so that all vertices in V 1 are on the northern boundary, all vertices in V 2 are on the southern boundary, all edges are within the square, and ....

Michael R. Garey and David S. Johnson. Crossing Number is NPComplete. SIAM J. Alg. Disc. Meth., 4:312--316, 1983.

....H is the smallest possible number of edge crossings in a drawing of H in the plane. A graph H is crossingcritical if cr(H e) cr(H) for all edges e 2 E(H) A graph H is k crossing critical if H is crossing critical and cr(H) k. Determining the crossing number of a graph is a hard problem [9] in general, and the crossing number is not even known exactly for complete or complete bipartite graphs. So it is important to study crossing critical graphs in order to understand what structural properties force the crossing number of a graph to be large. In this work, we prove that if G is a ....

M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983), 312-316.

....summarize selected results on the time complexity of some fundamental graph drawing problems. 9 Table 1.6: The time complexity of some fundamental graph drawing problems: general graphs and digraphs. Class of Graphs Problem Time Complexity Source general graph minimize crossings NP hard [54] 2 layered graph minimize crossings in layered drawing with preassigned order on one layer NP hard [43] general graph compute maximum planar subgraph NP hard [53] general graph planarity testing and computing a planar embedding O(n) Omega Gamma n) 8, 13, 47, 22, 68, 82] general graph ....

M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4(3):312--316, 1983.

....in the appropriate layer. We can now assume that we are given a layering such that no edge spans any layer. Note that the number of crossings is now dependent only on the ordering of vertices in each layer. Finding an ordering that minimizes the number of edge crossings is another NP hard problem [9]. A common heuristic is the layer by layer sweep, in which the ordering in, say, L 0 is xed and L 1 is reordered to reduce the number of crossings. Then, the order in L 1 is xed, and L 2 is reordered, and so on. After reaching L k , the process is reversed and repeated up and down the layering ....

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 Journal on Algebraic and Discrete Methods, 4:312-316, 1983.

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Gary, M. R. & Johnson, D. S. (1983), `Crossing number is NP-complete', SIAM J. Algeraic and Discrete Methods 4, 312 -- 316.

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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.R. Garey and D.S. Johnson, Crossing number is NP{complete, SIAM J. Alg. Disc. Meth. 1 (1983), 312-316. 6

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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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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, 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, D.S.JohuN"N Crossing number is NP-complete, SIAM J. Algebraic

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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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M. R. Garey and D. S. Johnson, "Crossing number is NP-complete", SIAM J. of Algebraic and Discrete Methods, 4 (1983) 312-316.

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