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Crossing number is NP-complete. (English) Zbl 0536.05016
The general crossing number decision problem is defined as follows: ”Given a graph G (multiple edges are allowed) and an integer k, is the crossing number of G less than or equal to k?” The authors prove that the crossing number decision problem is NP-complete, and hence likely to be intractable. By a simple argument it is shown that crossing number is in NP. To prove the NP-completeness, the known NP-complete problem of optimal linear arrangement (”Given a graph \(G=(V,E)\) and an integer k, is there a bijection \(f:V\to \{1,2,...,| V| \}\) such that \(\sum_{(u,v)\in E}| f(u)-f(v)| \leq k\)?”) is shown to be polynomially transformable to crossing number.
Reviewer: J.Širáň

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
68Q25 Analysis of algorithms and problem complexity
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