The crossing number of a graph @math is the least number of crossings over all possible drawings of @math . We present a structural characterization of graphs with crossing number one.
The problem of characterizing graphs with crossing number at least two was already studied by Arroyo and Richter @cite_0 in the context of .
@cite_0
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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. In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s-t paths, cycles, matchings, and @k-factors for @[email protected]? 1,2 . We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k^1^-^@e for any @e>0, where k is the number of crossings in G. We then give a simple fixed-parameter algorithm that tests in O^@?(2^k) time whether G has a crossing-free configuration for any of the above, where the O^@?-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O^@?(1.9999992^k) for spanning trees and O^@?((3)^k) for s-t paths and cycles. For spanning trees we also have an O^@?(1.968^k)-time Monte-Carlo algorithm. Each O^@?(@b^k)-time decision algorithm can be turned into an O^@?((@b+1)^k)-time optimization algorithm that computes a configuration with the minimum number of crossings. It is very well-known that there are precisely two minimal non-planar graphs: K 5 and K 3 , 3 (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple, 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the Mobius Ladder V 10 ; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V 10 ; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V 8 . Our main result includes the following, slightly surprising, fact: a 4-connected nonplanar graph G has crossing number at least 2 if and only if, for every pair e,f of edges having no common incident vertex, there are vertex-disjoint cycles in G with one containing e and the other containing f.