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In our paper, we construct a real-algebraic function whose Reeb(Kronrod-Reeb) graph is a graph respecting some algebraic domain: a graph forthis is called Poincar\'e-Reeb graph. The Reeb graph of a smooth function isdefined as a natural graph which is the quotient space of the manifold of thedomain under a natural equivalence relation for some wide and nice class ofsmooth functions. The vertex set is defined as the set of all connectedcomponents containing some singular points of the function: a singular point ofa smooth function is a point where the differential vanishes. Morse-Bottfunctions give very specific cases. The relation is  to contract each connected component of each preimage to a point.  Sharko has asked a natural and important problem: can we construct a nicesmooth function whose Reeb graph is a given graph? Explicit answers have beengiven first by Masumoto-Saeki in a generalized manner for closed surfaces.After that various answers have been presented by various researchers and mostof them are essentially for functions on closed surfaces and Morse functionssuch that connected components of preimages containing no singular points arespheres. Recently the author has also considered questions and solved in casesthe preimages are general manifolds.

Real algebraic functions on closed manifolds whose Reeb spaces are given graphs