Cop and Robber

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When the cop starts at a vertex and the robber is restricted to moves between vertices, this strategy also limits the cop to vertices, so it is a valid winning strategy for the visibility graph. In this game, one player controls the position of a given number of cops and the other player controls the position of a robber. The computability of algorithmic problems involving cop-win graphs has also been studied for infinite graphs. Arboricity, h-index, and dynamic algorithms", Theoretical Computer Science, 426–427: 75–90, arXiv: 1005. These numbers allow the algorithm to count, for any two vertices x and y, how much B contributes to the deficit of x and y, in constant time, by a combination of bitwise operations and table lookups.

On bridged graphs and cop-win graphs", Journal of Combinatorial Theory, Series B, 44 (1): 22–28, doi: 10. Following this strategy will result either in an actual win of the game, or in a position where the robber is on v and the cop is on the dominating vertex, from which the cop can win in one more move. What tactics have you learned that might be useful for other activities, such as sports and other wide games? A similar game with larger numbers of cops can be used to define the cop number of a graph, the smallest number of cops needed to win the game. In the case of infinite graphs, it is possible to construct computable countably infinite graphs, on which an omniscient robber could always evade any cop, but for which no algorithm can follow this strategy.

The hereditarily cop-win graphs are the same as the bridged graphs, graphs in which every cycle of length four or more has a shortcut, a pair of vertices closer in the graph than they are in the cycle. These are graphs defined from the vertices of a polygon, with an edge whenever two vertices can be connected by a line segment that does not pass outside the polygon. The Levi graphs (or incidence graphs) of finite projective planes have girth six and minimum degree Ω ( n ) {\displaystyle \Omega ({\sqrt {n}})} , so if true this bound would be the best possible. This is not true for all cop-win graphs; for instance, the five-vertex wheel graph is cop-win but contains an isometric 4-cycle, which is not cop-win, so this wheel graph is not hereditarily cop-win.

In graph theory, a cop-win graph is an undirected graph on which the pursuer (cop) can always win a pursuit–evasion game against a robber, with the players taking alternating turns in which they can choose to move along an edge of a graph or stay put, until the cop lands on the robber's vertex. If there is only one cop, the robber can move to a position two steps away from the cop, and always maintain the same distance after each move of the robber.

For other types of graphs, there may exist infinite cop-win graphs of that type even when there are no finite ones; for instance, this is true for the vertex-transitive graphs that are not complete graphs. However, if there are two cops, one can stay at one vertex and cause the robber and the other cop to play in the remaining path. To speed up its computations, Spinrad's algorithm uses a subroutine for counting neighbors among small blocks of log 2 n vertices. The 'Cops' should work together to trap 'Robbers' and defend the items, while 'Robbers' should also work together to distract the 'Cops' and get past them.