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Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaardsplitting in an orientable, irreducible 3-manifold M and DVW(F) the subset ofthe disk complex D(F) consisting of simplices having at least one vertex from Vand at least one vertex from W. In this article, we describe the shape ofDVW(F) and prove that there is a function from the components of DVW(F) to theisotopy classes of the generalized Heegaard splittings obtained by weakreductions from (V,W;F), where this function describes how the thick levels areembedded in the relevant compression bodies.

On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three I - the Structure Theorem