Abstract
The proof of Wagner's conjecture by Robertson and Seymour gives a finite description of any family of graphs which is closed under the minor ordering. This description is a finite set of minimal graphs not in the family; these graphs are called the obstructions of the family. Since the intersection and union of two minor closed graph families is again a minor closed graph family, an interesting question regards computing the obstructions of the new family given the obstructions for the original two families. It is easy to compute the obstructions of the intersection, but nontrivial to compute those of the union. In this paper, we show that if the original families are planar then the planar obstructions of the union are no larger than nO(n2), where n is the size of the largest obstruction of the original families.
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