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. Quilliot, Alain (1978), Jeux et pointes fixes sur les graphes [ Games and fixed points on graphs], Thèse de 3ème cycle (in French), Pierre and Marie Curie University, pp. 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, there exist infinite chordal graphs, and even infinite chordal graphs of diameter two, that are not cop-win.

Cop-win graphs can be defined by a pursuit–evasion game in which two players, a cop and a robber, are positioned at different initial vertices of a given undirected graph.Create blocks from an arbitrary partition of the vertices, and find the numbers representing the neighbors of each vertex in each block. 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. 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. 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.

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.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. Construct a block of the log n removed vertices and numbers representing all other vertices' adjacencies within this block. 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} .