Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. II

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema and if a point is not a local extremum, then there exist its neighborhood and a number n ∈ ℕ such that the function restricted to this neighborhood is topologically conjugate to Re z ⁿ in a certain neighborhood of zero. Given f : M ² → ℝ, let Γ _K−R (f) be a quotient space of M ² with respect to its partition formed by the components of level sets of the function f. It is known that the space Γ _K−R (f) is a topological graph if M ² is compact. In the first part of the paper, we introduced the notion of graph with stalks that generalizes the notion of topological graph. For noncompact M ² , we present three conditions sufficient for Γ _K−R (f) to be a graph with stalks. In the second part, we prove that these conditions are also necessary in the case M ² = ℝ² . In the general case, one of our conditions is not necessary. We provide an appropriate example.