Crossing Number is Hard for Kernelization

The graph crossing number problem, cr(G) ≤ k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixedparameter tractable for the parameter k [Grohe]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G) ≤ k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge. 1998 ACM Subject Classification F.2.2 [Nonnumerical Algorithms and Problems] Geometrical problems and computations, F.1.3 [Complexity Measures and Classes] Reducibility and completeness