On such graphs, every algorithm for choosing moves for the cop can be evaded indefinitely by the robber. 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.

And in all other cases, the cop follows the edge in H that is the image under the retraction of a winning edge in G. The cop can start anywhere, and at each step move to the unique neighbor that is closer to the robber.a b c d e f g h i Nowakowski, Richard; Winkler, Peter (1983), "Vertex-to-vertex pursuit in a graph", Discrete Mathematics, 43 (2–3): 235–239, doi: 10. Language localizations: English, Chinese, Japanese, Russian, French, German, Korean, Spanish, Portuguese.

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. 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. Two types of subgraph that are guardable are the closed neighborhood of a single vertex, and a shortest path between any two vertices.

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.