Cop and Robber

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If B is a set of vertices that the algorithm has selected to be a block, then for any other vertex, the set of neighbors of that vertex in B can be represented as a binary number with log 2 n bits.

One of the two kings, playing as cop, can beat the other king, playing as robber, on a chessboard, so the king's graph is a cop-win graph.Chepoi, Victor (1997), "Bridged graphs are cop-win graphs: an algorithmic proof", Journal of Combinatorial Theory, Series B, 69 (1): 97–100, doi: 10. Analogously, it is possible to construct computable countably infinite cop-win graphs, on which an omniscient cop has a winning strategy that always terminates in a finite number of moves, but for which no algorithm can follow this strategy. Cops N Robbers (FPS) is a 3d pixel style online multiplayer gun shooting games with gun craft feature. On bridged graphs and cop-win graphs", Journal of Combinatorial Theory, Series B, 44 (1): 22–28, doi: 10. Henri Meyniel (also known for Meyniel graphs) conjectured in 1985 that every connected n {\displaystyle n} -vertex graph has cop number O ( n ) {\displaystyle O({\sqrt {n}})} .

One way to prove this is to use subgraphs that are guardable by a single cop: the cop can move to track the robber in such a way that, if the robber ever moves into the subgraph, the cop can immediately capture the robber.

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. A cop following this inductive strategy on a graph with n vertices takes at most n moves to win, regardless of starting position. 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. The product-based strategy for the cop would be to first move to the same row as the robber, and then move towards the column of the robber while in each step remaining on the same row as the robber.