The game of Cops and Eternal Robbers

The one-visibility localization game

“…Section 5 focuses on Cartesian grids, in which Theorem 18 gives the exact value of ζ k for such graphs in many cases, with the remaining cases being one of two values as long as n is sufficiently large. This result also separates the parameters, showing that for all i < j, there exists a graph G with ζ i (G) ≠ ζ j (G), settling an open problem stated in [9]. We finish with open problems.…”

“…We denote the minimum number of cops required to win the k-proximity game on a graph G as prox k (G). This is a generalization of the blind localization game first considered in [11], which was later reintroduced as the proximity game in [9]. The k-proximity game is also analogous to the k-visibility seeing cop-number introduced in [15].…”

“…The domination number of G, written γ(G), is the minimum cardinality of a dominating set. In [9], the authors prove that if G is C 4 -free, ζ 1 (G) ≤ γ(G) + ∆(G). To generalize their result, consider the size of a minimal j-dominating set, written γ j (G), which is a set S with the property that for each vertex x ∉ S, there is some vertex in S at a distance at most j from x.…”

“…The game in the present form was first considered in the paper [14], and subsequently studied in several papers such as [1,5,6,7,11,12,13,21]. We consider a version of limited visibility Cops and Robbers in the setting of the Localization game, first introduced in [9]. Let k be a non-negative integer.…”

“…A recent work [2] introduced the so-called zero-visibility search game, which is equivalent to ζ k when k = 0. The case k = 1 was studied in [9] and the k-visibility Localization was studied for random graphs in [22].…”