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

For a smooth function on a smooth manifold of a suitable class, the space of all connected components of preimages is the graph and called the {\it Reeb graph}. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions which are not so wild. In this paper, we study whether we can construct a smooth function with good geometric properties inducing a given graph as the Reeb graph. This problem has been essentially launched by Sharko in 2000s and various answers have been given by Masumoto, Michalak, Saeki, and so on. Recently the author set a new explicit problem and gave an answer. In the studies before the result of the author, considered functions are smooth functions on closed surfaces or Morse functions such that preimages of regular values are disjoint unions of standard spheres. On the other hand, the author constructed a smooth function on a suitable $3$-dimensional closed, connected and orientable manifold inducing the Reeb graph isomorphic to the given graph such that preimages of regular values are arbitrary closed surfaces. Based on this result and method of the author, with several new ideas, we will consider smooth functions on surfaces and manifolds which may be non-closed and give answers to the problem.