Crossing number is hard for cubic graphs☆

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Abstract

It was proved by [M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983) 312–316] that computing the crossing number of a graph is an NP-hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NP-hard to determine the crossing number of a simple 3-connected cubic graph. In particular, this implies that the minor-monotone version of the crossing number problem is also NP-hard, which has been open till now.

Keywords

Crossing number
Cubic graph
NP-completeness

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An extended abstract published in: Proceedings MFCS'04, in: Lecture Notes in Comput. Sci., vol. 3153, Springer, 2004, pp. 772–782.

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Supported by Czech research grant GAČR 201/05/0050.

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