# Some provably hard crossing number problems

@article{Bienstock1990SomePH, title={Some provably hard crossing number problems}, author={Daniel Bienstock}, journal={Discrete \& Computational Geometry}, year={1990}, volume={6}, pages={443-459} }

This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists. In particular, we show that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph. Using some recent results of Goodman, Pollack, and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of…

## 102 Citations

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A nontrivial derandomization of the natural randomized 1/3-approximation algorithm is given, which generalizes to a weighted setting as well as to an ordering constraint satisfaction problem.

### Which crossing number is it, anyway? [computational geometry]

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- 1998

It is proved that the determination of each of these parameters is an NP-complete problem and that the largest of these numbers cannot exceed twice the square of the smallest (the odd-crossing number).

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

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### Computing crossing number in linear time

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We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph in…

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It is shown that the crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3, and an NP-hardness proof of the weighted crossing number already for pathwidth 3 graphs and bicliques is provided.

### Which Crossing Number Is It Anyway?

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- 1998

A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the…

### The Complexity of Drawing Graphs on Few Lines and Few Planes

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This work investigates the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes, and shows lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes.

### The Rectilinear Crossing Number of K10 is 62

- MathematicsElectron. J. Comb.
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A purely combinatorial argument is used to show that the rectilinear crossing number of K_10 is 62, and an asymptotic lower bound for a related problem is improved.

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This thesis deals with the rectilinear crossing minimization problem, which is NP-hard [BD93]. More precisely, we propose a heuristic for computing a straight-line drawing of a general graph G which…

### Which Crossing Number Is It Anyway?

- MathematicsJ. Comb. Theory, Ser. B
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It is proved that the largest of these numbers (the crossing number) cannot exceed, twice the square of the smallest (the odd-crossing number) and the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-Crossing number.

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