On some problems of guaranteed search on graphs

Pursuit-evasion differential games on graphs with no information on the evader are considered. Special attention is given to the following ɛ-search problem posed by Golovach. A topological graph embedded in three-dimensional Euclidean space is considered. For simplicity, its edges are assumed to be polygonal, only simple motions of the pursuers and the evader are allowed, and the graph is a phase constraint for all players. The goal of the pursuers is to construct a program depending only on the structure of the graph which ensures capturing the invisible evader, i.e., approaching the evader at a distance of at most ɛ (in the intrinsic metric of the graph), where ɛ is a given nonnegative number. The problem consists in finding the minimum number of pursuers (called the ɛ-search number) needed to capture the evader. Properties of the Golovach function, which assigns the ɛ-search number to every nonnegative ɛ, are investigated. Golovach and Petrov proved that the Golovach function for the complete graph on more than five vertices may have non-unit jumps. The authors of this paper are aware of examples of similar degeneration for trees. These examples disprove the conjecture that the Golovach function of any planar graph has only unit jumps. A subclass of trees for which the Golovach function has only unit jumps is distinguished.