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The present paper is on a new result on so-called realization problems ofgraphs as Reeb graphs of Morse functions with prescribed preimages. The Reebgraph of a smooth function is a graph obtained by identifying two points in themanifold of the domain if and only if they are in a same connected component ofsome preimage. In considerable cases such as cases where functions areMorse-Bott functions on closed manifolds we obtain such graphs.  Reeb graphs represent the manifolds of the domains compactly. They are notonly fundamental tools in geometry using differentiable functions and maps asfundamental tools, but also in applications of mathematics such asvisualizations. Realization problems are also fundamental and important.Construction of functions with prescribed preimages was considered first by theauthor essentially and before the present study, the author has obtained aresult.

Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages