A Cop and Drunken Robber Game on n-DimensionalInfinite-Grid Graphs

A Cop and Drunken Robber (CDR) game is one variation of a famous combinatorial game, called Cops and Robbers, which has been extensively studied and applied in the area of theoretical and computer science as demonstrated by several conferences and publications. In this paper, for a natural number n, we present two strategies for a single cop to chase a drunken robber on n-dimensional infinite-grid graphs. Both strategies show that if the initial distance between the cop and the drunken robber is s, then the expected capture time is s+o(s).

Cops, robbers, and barricades

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

Cops, robbers, and barricades

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

Cops and robbers on graphs and hypergraphs

Cops and Robbers is a vertex-pursuit game played on a graph where a set of cops attempts to capture a robber. Meyniel's Conjecture gives as asymptotic upper bound on the cop number, the number of cops required to win on a connected graph. The incidence graphs of affine planes meet this bound from below, they are called Meyniel extremal. The new parameters mқ and Mқ describe the minimum orders of k-cop-win graphs. The relation of these parameters to Meyniel's Conjecture is discussed. Further, the cop number for all connected graphs of order 10 or less is given. Finally, it is shown that cop win hypergraphs, a generalization of graphs, cannot be characterized in terms of retractions in the same manner as cop win graphs. This thesis presents some small steps towards a solution to Meyniel's Conjecture.

The CC-Game: A Variant Of The Game Of Cops And Robbers

Cops and Robbers is a vertex pursuit game played on graphs. The objective of the game, as the name suggests, is for a set of cops to catch the robber. We study a new variant of this game in which the robber can attack a cop or fight back. This variation restricts the movement of the cops and changes many of the parameters and strategies achieved in the regular game. We explore aspects of this variant such as classifications for certain cop numbers, upper and lower bounds, strategies on special graphs, the cop number on products of graphs, complexity of computations, and density of cops in infinite graphs.