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For a smooth function on a smooth manifold of a suitable class, the space ofall connected components of preimages is the graph and called the {\it Reebgraph}. Reeb graphs are fundamental tools in the algebraic and differentialtopological theory of Morse functions and more general functions which are notso wild. In this paper, we study whether we can construct a smooth functionwith good geometric properties inducing a given graph as the Reeb graph. Thisproblem has been essentially launched by Sharko in 2000s and various answershave been given by Masumoto, Michalak, Saeki, and so on. Recently the authorset a new explicit problem and gave an answer. In the studies before the resultof the author, considered functions are smooth functions on closed surfaces orMorse functions such that preimages of regular values are disjoint unions ofstandard spheres. On the other hand, the author constructed a smooth functionon a suitable $3$-dimensional closed, connected and orientable manifoldinducing the Reeb graph isomorphic to the given graph such that preimages ofregular values are arbitrary closed surfaces. Based on this result and methodof the author, with several new ideas, we will consider smooth functions onsurfaces and manifolds which may be non-closed and give answers to the problem.

On Reeb graphs induced from smooth functions on closed or open manifolds