Surrounding cops and robbers on graphs of bounded genus

Abstract

We consider a surrounding variant of cops and robbers on graphs of bounded genus. We obtain bounds on the number of cops required to surround a robber on planar graphs, toroidal graphs, and outerplanar graphs. We also obtain improved bounds for bipartite planar and toroidal graphs. We briefly consider general graphs of bounded genus and graphs with a forbidden minor.Definition 2.1. Let G be a graph. Let P be a (u, v)-path of length l in G. We say that P is geodesic with respect to G if all (u, v)-paths in G have length at least l.Definition 2.2. Let G be a graph with a subgraph H. We say that H is geodesically closed with respect towhich contradicts the assumption that P is a geodesic path.The following lemma shows that we may guard a geodesic path with three cops.Lemma 2.4. Let G be a graph. Let (v 0 , . . . , v k ) be a geodesic path of G. Then there exists a strategy using three cops such that after a finite number of moves, the robber is unable to access P .Proof. We name our cops C 1 , C 2 , C 3 . Whenever the robber occupies a vertex w ∈ G with dist(w, v 0 ) = d such that d ≤ k, we say that the robber has a shadow at the vertex v d . Whenever the robber occupies a vertex w ∈ G with dist(w, v 0 ) ≥ k, we say that the robber has a shadow at v k . When the robber moves, the distance from the robber to v 0 changes by at most 1, and thus after each robber move, the robber's shadow on P either stays put or moves to an adjacent vertex of P . Therefore, after a finite number of moves, we can reach one of the following types of positions:

Cops and an Insightful Robber

“…The cop number of a planar graph is at most 4. For the surrounding model the upper bound is 6 [4]. Since the icosahedron is regular of degree 5 and, in general, the average degree of a planar graph is strictly less than 6, then a lower bound is for planar graphs is 5.…”

“…Closely related game variants to the Cheating Robot are: (1) simultaneously moving cops and robbers [11] and (2) surrounding cops and robbers (preprints [5] and [4]). The former investigates a variant where players are all moving simultaneously.…”

Primal-Dual Cops and Robber

“…In these variants the robber is captured if every adjacent vertex, respectively every incident edge, is occupied by a cop. The smallest number of cops that always suffices for any planar graph G is 3 in the classical variant [1], 7 in the surrounding variant [4], 7∆(G) in the containment variant [6] and 3 when both, cops and robber, move on edges [7].…”