Our contribution is twofold: (1) We formulate and solve a particular graph-theoretic problem: For an undirected graph G admitting a so-called MB-good orientation (acyclic with all its sources and sinks having degree at most 2), we calculate the minimum number \(\nu (G)\) of “transit” vertices (neither sources nor sinks) of degree 2 over all MB-good orientations of G. Surprisingly, \(\nu (G)\le 1\) , and for most graphs, there exist MB-good orientations with all vertices of degree 2 being sources or sinks, i.e., with \(\nu (G)=0\) . As an application, (2) in the context of the Reeb graph theory, we characterize the set of Reeb graphs defined by all Morse–Bott functions on a given surface; namely, we prove a criterion on realization of a graph as the Reeb graph of a Morse–Bott function on this surface.
