Cop and Robber

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The cops are trying to catch the robber by moving to the same position, while the robber is trying to remain uncaught.

This can be proved by mathematical induction, with a one-vertex graph (trivially won by the cop) as a base case.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. 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. 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.

The cop number of a graph G {\displaystyle G} is the minimum number k {\displaystyle k} such that k {\displaystyle k} cops can win the game on G {\displaystyle G} .Chepoi, Victor (1997), "Bridged graphs are cop-win graphs: an algorithmic proof", Journal of Combinatorial Theory, Series B, 69 (1): 97–100, doi: 10. On bridged graphs and cop-win graphs", Journal of Combinatorial Theory, Series B, 44 (1): 22–28, doi: 10. A family of mathematical objects is said to be closed under a set of operations if combining members of the family always produces another member of that family. However, there exist infinite chordal graphs, and even infinite chordal graphs of diameter two, that are not cop-win. Language localizations: English, Chinese, Japanese, Russian, French, German, Korean, Spanish, Portuguese.

Then, while staying in pairs whose first component is the same as the robber, the cop can play to win in the second of the two factors. 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. However, the problems of obtaining a tight bound, and of proving or disproving Meyniel's conjecture, remain unsolved.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. Conversely, almost all dismantlable graphs have a universal vertex, in the sense that, among all n-vertex dismantlable graphs, the fraction of these graphs that have a universal vertex goes to one in the limit as n goes to infinity. Even when the cop and robber are allowed to move on straight line segments within the polygon, rather than vertex-to-vertex, the cop can win by always moving on the first step of a shortest path to the robber. Create blocks from an arbitrary partition of the vertices, and find the numbers representing the neighbors of each vertex in each block. Bonato and Nowakowski describe this game intuitively as the number of ghosts that would be needed to force a Pac-Man player to lose, using the given graph as the field of play.

A cop-win graph is hereditarily cop-win if and only if it has neither the 4-cycle nor 5-cycle as induced cycles. The game with a single cop, and the cop-win graphs defined from it, were introduced by Quilliot (1978). On such graphs, every algorithm for choosing moves for the cop can be evaded indefinitely by the robber.In this game, one player controls the position of a given number of cops and the other player controls the position of a robber. It constructs and maintains the actual deficit set (neighbors of x that are not neighbors of y) only for pairs ( x, y) for which the deficit is small. Construct a block of the log n removed vertices and numbers representing all other vertices' adjacencies within this block. Arboricity, h-index, and dynamic algorithms", Theoretical Computer Science, 426–427: 75–90, arXiv: 1005. 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.