A New Variant of the Localization Game

In the Localization game, a set of cops use distance probes to attempt to capture an invisible robber on a graph. The present work introduces a variant in which these probes are restricted to reveal only distances of 0 or 1. This is explored for general graphs, planar graphs, and extensively for trees, on which the variant is drastically different from the original game. A characterization is given of all graphs on which a single cop can win the modified game. We also introduce a relaxed version of the variant, which is found to be particularly useful in the search for bounds.