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 a a"center dot 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) -> a"e, let I" (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 I" (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 I" (K-R) (f) to be a graph with stalks.