Instead, every algorithm for choosing moves for the robber can be beaten by a cop who simply walks in the tree along the unique path towards the robber. Create blocks from an arbitrary partition of the vertices, and find the numbers representing the neighbors of each vertex in each block. By choosing the cop's starting position carefully, one can use the same idea to prove that, in an n-vertex graph, the cop can force a win in at most n − 4 moves. By Kőnig's lemma, such a tree must have an infinite path, and an omniscient robber can win by walking away from the cop along this path, but the path cannot be found by an algorithm. A cop following this inductive strategy on a graph with n vertices takes at most n moves to win, regardless of starting position.

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. Explain the rules of the game clearly and have a clear way to communicate that the game must stop when needed.

Make sure everyone understands what contact is acceptable, and monitor contact throughout the activity. Repeatedly find a vertex v that is an endpoint of an edge participating in a number of triangles equal to the degree of v minus one, delete v, and decrement the triangles per edge of each remaining edge that formed a triangle with v. The game with a single cop, and the cop-win graphs defined from it, were introduced by Quilliot (1978). 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. Each of the cop's steps reduces the size of the subtree that the robber is confined to, so the game eventually ends.

In graph theory, a branch of mathematics, the cop number or copnumber of an undirected graph is the minimum number of cops that suffices to ensure a win (i. Additionally, if v is a dominated vertex in a cop-win graph, then removing v must produce another cop-win graph, for otherwise the robber could play within that subgraph, pretending that the cop is on the vertex that dominates v whenever the cop is actually on v, and never get caught.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. This can be proved by mathematical induction, with a one-vertex graph (trivially won by the cop) as a base case. Download Cops N Robbers(FPS) for a great online multiplayer pixel gun shooting game experience, whether you are a fps games or block building games fan! The computability of algorithmic problems involving cop-win graphs has also been studied for infinite graphs. 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.