Some provably hard crossing number problems

  title={Some provably hard crossing number problems},
  author={Daniel Bienstock},
  journal={Discrete \& Computational Geometry},
  • D. Bienstock
  • Published 1 May 1990
  • Mathematics
  • Discrete & Computational Geometry
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… 

Approximating the Maximum Rectilinear Crossing Number

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]

  • J. PachG. Tóth
  • Mathematics
    Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
  • 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).

Computing crossing number in linear time

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

Crossing Number for Graphs with Bounded Pathwidth

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?

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

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

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.

Rectilinear Crossing Minimization Master Thesis of Klara Reichard

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?

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.



Rectilinear drawings of graphs

A complete characterization of the pairs of dual graphs that can be represented as geometric dual graphs such that all edges except one are straight line segments is obtained.

Crossing Number is NP-Complete

This paper shows that there is not likely to be any efficient way to design an optimal embedding of a graph or network in a planar surface, and hence this problem is NP-complete.

Small sets supporting fary embeddings of planar graphs

It is shown that any set F, which can support a Fáry embedding of every planar graph of size n, has cardinality at least n, which settles a problem of Mohar.

Upper bounds for configurations and polytopes inRd

It follows as a corollary of the first result that there are no more thannd(d+1)n combinatorially distinct labeled simplicial polytopes inRd withn vertices, which improves the best previous upper bound ofncnd/2.

Coordinate representation of order types requires exponential storage

We give doubly exponential upper and lower bounds on the size of the smallest grid on which we can embed every planar configuration of n points in general position up to order type. The lower bound

Proof of Grünbaum's Conjecture on the Stretchability of Certain Arrangements of Pseudolines

Semispaces of Configurations

A non-vertical hyperplane h, disjoint from a finite set P of points, partitions P into two sets P+ = P∩h+ and P- = P∩h - called semispaces of P, where h+ is the open half-space above hyperplane h and

Embedding planar graphs on the grid

We show that each plane graph of order n 2 3 has a straight line embedding on the n-2 by n-2 grid. This embedding is computable in time O(n). A nice feature of the vertex-coordinates is that they

Proof of a conjecture of Burr, Grünbaum, and Sloane