  The Reeb space of a function or a map on a manifold is defined as the space
of all connected components of preimages and represents the manifold compactly.
In fact, Reeb spaces are fundamental and useful tools in geometric theory of
so-called Morse functions and more general maps which are sufficiently tame.
  Can we construct an explicit good function inducing a given graph as the Reeb
space (Reeb graph)? These problems were launched by Sharko in 2000s and have
been explicitly solved by several researchers. As related pioneering studies,
the author also found and solved problems adding constraints on singularities
and preimages for example.
  The present paper concerns new problems on these works. We define the classes
of maps onto graphs locally regarded as ones onto the Reeb spaces induced from
smooth functions of suitable classes and consider and challenge the problems
for the classes.
