Defining Grizzly Bear Graphs. We represent the house by a graph. ... you get to place yourself and the grizzly bears on the graph, wherever you want. ...
Jim Belk and Maria Belk
can escape from grizzly bears?
Let’s make this more precise.
There is a similar well-known game:
The difference between the two games:
The zombie can catch the person:
The cop cannot catch the robber:
The cop number of a graph , denoted , is the minimum number of cops needed to eventually catch the robber, assuming the robber uses the best possible strategy.
Theorem. (Seymour and Thomas) The cop number of a graph equals the treewidth plus 1.
Theorem. The zombie number of a graph is either or .
The following graph has cop number 3 and zombie number 2:
If there are only 2 zombies, you can always move to whichever of the three vertices is the furthest from both zombies.
The following graph has cop number 3 and zombie number 3.
A graph with cop number 3:
Theorem. The “minimal” graphs with zombie number 3 are the following:
A graph has zombie number 2 if does not contain one of the above graphs as a minor.