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The Reeb graph of a function on a smooth manifold is the graph obtained asthe space of all connected components of level sets such that the set of allvertices coincides with the set of all connected components of level setsincluding singular points. Reeb graphs are fundamental and important in thealgebraic and differential topological theory of Morse functions and theirgeneralizations.  In this paper, as a related fundamental and important study, for givengraphs, we construct certain smooth functions inducing the graphs as the Reebgraphs. Such works have been demonstrated by Masumoto, Michalak, Saeki, Sharkoetc. and also by the author since 2000s. We present new smooth functions onsuitable $3$-dimensional closed orientable manifolds through explicitconstructive methods.

On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values