The Game of Cops and Robbers on Graphs: How Many Cops Do You Need?

👮The cop number of a graph is the minimum number of cops required to guarantee capture of the robber.

🕵️A graph is cop-win if the cop has a strategy to guarantee capture, and robber-win if the robber has a strategy to avoid capture.

🌲The cop number of a tree is 1, meaning a single cop can guarantee capture of the robber.

🔺The cop number of a cycle is 2, as two cops can trap the robber between them.

🔲The cop number of a dodecahedron is 3, according to a result that bounds the cop number of planar graphs.

What is the cop number of a graph?

—The cop number of a graph is the minimum number of cops required to guarantee capture of the robber in the game of Cops and Robbers.

What does it mean for a graph to be cop-win?

—A graph is cop-win if the cop has a strategy to guarantee capture of the robber in the game of Cops and Robbers.

What does it mean for a graph to be robber-win?

—A graph is robber-win if the robber has a strategy to avoid capture by the cop in the game of Cops and Robbers.

What is the cop number of a tree?

—The cop number of a tree is 1, meaning a single cop can guarantee capture of the robber in the game of Cops and Robbers.

What is the cop number of a cycle?

—The cop number of a cycle is 2, as two cops can trap the robber between them in the game of Cops and Robbers.

00:00The game of Cops and Robbers on graphs is explored, where the cop tries to catch the robber by moving along edges.

01:30The cop number of a graph is the minimum number of cops required to guarantee capture of the robber.

03:02The cop number of a cycle is 2, as two cops can trap the robber between them.

05:41The cop number of a dodecahedron is 3, according to a result that bounds the cop number of planar graphs.

08:16Meyniel's conjecture states that the cop number of a graph grows like the square root of the number of vertices.