Making Your House Safe From Zombie Attacks. Jim Belk and Maria Belk. How can we construct a house so that we can escape from grizzly bears? Let’s make this more precise. Defining Grizzly Bear Graphs. We represent the house by a graph. Defining Grizzly Bear Graphs.
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.