Graph Searching Games with a Radius of Capture

The problem of guaranteed search on graphs, the so-called $\varepsilon$-search problem, is considered. The properties of the Golovach function which is the $\varepsilon$-search number as the function of the radius of capture $\varepsilon$ are studied. In the present work, we show that the jumps of the Golovach function for trees may have an arbitrarily large height. We describe some classes of trees with the non-degenerate Golovach function, and construct the minimal tree with a non-unit jump.