Arbitrarily large jumps of the Golovach function for trees

Abstract

The ɛ-search problem on connected topological graphs is considered. The jumps of the Golovach function are studied for trees. As is known, the Golovach function for trees with at most 27 edges has only unit jumps. In the authors’ earlier papers, examples of trees on which the Golovach function has a jump of size 2 were constructed. In the present paper, it is shown that the jumps of the Golovach function for trees may be arbitrarily large. A sharp bound for the size of jumps is given for a sequence of trees constructed in the paper. A theorem about small perturbations of edge lengths for trees is proved, which asserts an arbitrarily small perturbation of the edge lengths of a given tree (whose Golovach function may be degenerate) may yield a new tree whose Golovach function has only unit jumps.