Localization game on geometric and planar graphs

Bounds on the localization number

“…In particular, under our model, if ζ ( G ) = 1, then χ ( G ) ≤ 3. Bosek et al [5] asked whether χ ( G ) is, in general, bounded above by some function of ζ ( G ). We answer this question in the affirmative; Theorem 2.1 yields a short proof.…”

“…Bosek et al [5] showed that ζ ( G ) can be unbounded on the class of planar graphs and asked whether the same is true of outerplanar graphs. They answer this question in the negative in [6], by showing that ζ ( G ) ≤ 3 when G is outerplanar.…”

“…The bound of , where Δ is the maximum degree of G , was shown in [16]. In [5], Bosek et al showed that ζ ( G ) is bounded above by the pathwidth of G and that the localization number is unbounded even on graphs obtained by adding a universal vertex to a tree. They also proved that computing ζ ( G ) is NP ‐hard for graphs with diameter 2, and they studied the localization game for geometric graphs.…”

“…Bosek et al conjectured (see [5], Conjecture 16) that there is a function f such that every graph with ζ ( G ) ≤ k satisfies χ ( G ) ≤ f ( k ), where χ ( G ) is the chromatic number of G . We settle this conjecture in Corollary 2.2.…”

A note on the localization number of random graphs: Diameter two case

“…, k is sufficient to determine v. Otherwise, the robber will move to a neighbor of v and the cop will try again with a (possibly) different test set W . Given G, the localization number ζ(G) is the minimum k so that the cop can eventually locate the robber, that means, the cop determines the exact location of the robber from the test sets of size k. This game was introduced by Bosek et al [3], who studied the localization game on geometric and planar graphs, and also independently, by Haslegrave et al [6]. For some other related results see [4,8,9].…”