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We give a new answer to so-called realization problems of graphs as Reebgraphs of smooth functions with prescribed preimages of regular values havingnice structures. We present a best possible answer for functions on3-dimensional closed manifolds.  The Reeb space of a smooth function is the quotient space of the manifold ofthe domain induced from the following equivalence relation; two points in themanifold are equivalent if and only if they are points of a same connectedcomponent of some preimage. They are in considerable cases graphs (Reebgraphs).  Reeb spaces with preimages represent the manifolds well and are importanttools in geometry. Recently they play important roles in applications ofmathematics such as visualizations. Realization problems ask us whether we canconstruct smooth functions with prescribed Reeb graphs and preimages. Studieson construction respecting preimages were essentially started by the author.

Smooth functions with simple structures on 3-dimensional closed manifolds with prescribed Reeb graphs and preimages