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

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema; if a point is not a local extremum, then there exist its neighborhood and a number n ∈ ℕ such that a 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 the level sets of f. It is known that, for compact M ², the space Γ _K−R (f) is a topological graph. We introduce the notion of graph with stalks, which generalizes the notion of topological graph. For noncompact M ², we establish three conditions sufficient for Γ _K−R (f) to be a graph with stalks.