The realizability problem for Golovach-type functions

Abstract

The problem mentioned in the title is stated as follows. Consider a function f with some necessary properties of the Golovach function, namely, a piecewise constant nonincreasing right continuous function defined on the set of nonnegative real numbers and taking integer values such that this function is identically equal to 1 at sufficiently large argument values. The problem of realizing the function f in a class \(\mathbb{G}\) of topological graphs is to find a graph \(G \in \mathbb{G}\) such that its Golovach functions coincides with f. Examples of realization of some functions possessing the properties mentioned above are considered. In the simplest case, all graphs for which the function can be realized are described. For less trivial examples, realizability criteria for functions with the properties of the Golovach function in the class of trees and in the class of trees with given edge search number which have the least number of edges are presented.