This essay will present the vertex pursuit game of cops and robbers and the problem that made it famous: Meinyel's conjecture. The conjecture stood unproved from 1987 until 2010 when Łuczak and Prałat proved the conjecture with their "Zig-Zag Theorem", which is also covered in the essay.

Cops and Robbers on Graphs (C & R) is a vertex-to-vertex pursuit game played on graphs first introduced by Quilliot (in 1978) and Nowakowski (in 1983). The cop player starts the game by choosing a set of vertices which will be the cops’ starting positions. The robber player responds by choosing its own start vertex. On each player’s turn, the player may move its tokens to adjacent vertices. The cops win if the robber is captured (they occupy the same vertex). The robber wins if it can avoid capture indefinitely. The question, then, is to determine the smallest number of cops required to guarantee the robber will be captured. A variation of C & R called Zombies and Survivors (Z & S) was recently proposed and studied by Fitzpatrick. Z & S is the same as C & R with the added twist that the zombies are required to move closer to the survivor (by following a shortest path from the zombie to the survivor). Whenever multiple shortest paths exist, the zombies are free to choose which one to follow. As in C & R, we are interested in the minimum number of zombies required to guarantee the survivor will be caught. Chapter 1 summarizes important results in vertex-pursuit games. In Chapter 2 we give an example of a planar graph where 3 zombies always lose, whereas Aigner and Fromme showed in 1984 that three cops have a winning strategy on planar graphs. In Chapter 3 we show how two zombies can win on a cycle with one chord.