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**1 - 5**of**5**### LAZY COPS AND ROBBERS ON HYPERCUBES

"... Abstract. We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojak ..."

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Abstract. We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes. 1.

### A PROBABILISTIC VERSION OF THE GAME OF ZOMBIES AND SURVIVORS ON GRAPHS

"... Abstract. We consider a new probabilistic graph searching game played on graphs, in-spired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the c ..."

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Abstract. We consider a new probabilistic graph searching game played on graphs, in-spired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is strictly greater than 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids. 1.

### LAZY COPS AND ROBBERS PLAYED ON RANDOM GRAPHS AND GRAPHS ON SURFACES

"... We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. The lazy cop number is the analogue of the usual cop number for this game. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic up ..."

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We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. The lazy cop number is the analogue of the usual cop number for this game. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the analogue of the cop number of the hypercube. By investigating expansion properties, we provide asymptotically almost sure bounds on the lazy cop number of binomial random graphs G(n, p) for a wide range of p = p(n). We provide an upper bound for the lazy cop number of graphs with genus g by using the Gilbert-Hutchinson-Tarjan separator theorem.

### LAZY COPS AND ROBBERS PLAYED ON GRAPHS

"... We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who p ..."

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We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By investigating expansion properties, we provide asymptotically almost sure bounds on the lazy cop number of binomial random graphs G(n, p) for a wide range of p = p(n). By coupling the probabilistic method with a potential function argument, we also improve on the existing lower bounds for the lazy cop number of hypercubes. Finally, we provide an upper bound for the lazy cop number of graphs with genus g by using the Gilbert-Hutchinson-Tarjan separator theorem.

### Correction: Chasing a Fast Robber on Planar Graphs and Random Graphs

, 2015

"... Consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c∞(G) denote the number of cops needed to capture the robber in a graph G in this varia ..."

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Consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c∞(G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G. In [1, Theorem 1] we showed that if G is planar then c∞(G) = Θ(tw(G)), and there is a polynomial-time constant-factor approximation algorithm for computing c∞(G). One part of the argument, namely the proof of c∞(G) = Ω(tw(G)), was incomplete. Here we give a complete proof for this statement. We will need a few definitions. An apex graph is a graph H that has a vertex v such that H − v is planar. For two undirected simple graphs G and H, we say G contains H as a contraction if H can be obtained by applying a sequence of edge contractions to G. We say G is H-minor-free if H is not a subgraph of any contraction of G. For any positive integer r, let Γr be the graph obtained from the triangulated r×r grid by joining a degree-2 corner to all the boundary vertices. See Figure 1 for an illustration. Formally, we have V (Γr) = {(i, j) : 1 ≤ i, j ≤ r},