... The Quadrangulation Conjecture concerns the problem of finding a natural bijection of this type. Tutte's medial construction is a solution in the special case g = 0 of planar maps. We give a construction of a bijection e\Xi which both extends Tutte's medial construction to non-planar maps and preserves the parameter n of the Quadrangulation Conjecture. (The parameter g is not generally preserved, excep
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