Bounds on the localization number

06/13/2018
by   Anthony Bonato, et al.
0

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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset