Bounds on the localization number
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We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written ζ (G). We settle a conjecture of nisse1 by providing an upper bound on the localization number as a function of the chromatic number. In particular, we show that every graph with ζ (G) < k has degeneracy less than 3^k and, consequently, satisfies χ(G) < 3^ζ (G). We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.
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