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We study complete minimal graphs in IH × IR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in IH. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in IH × IR that are conformally the complex plane l C. The vertical projection of such a graph yields a harmonicdoi:10.4007/annals.2010.172.1879 fatcat:t6wuszmo7bfszevcbzui6ezgdq