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

• We represent the house by a graph. Vertices represent rooms.

Defining Grizzly Bear Graphs

• We represent the house by a graph. Vertices represent rooms, and edges represent hallways.

Defining Grizzly Bear Graphs

• We will allow loops and multiple edges in our graphs.

• There is no exit from the house.• At the start of the game, you get to place

yourself and the grizzly bears on the graph, wherever you want.

Defining Grizzly Bear Graphs

• You move much, much faster than the grizzly bears.

Defining Zombie Graphs

• You move much, much faster than the grizzly bears zombies.

Defining Zombie Graphs

• You move much, much faster than the grizzly bears zombies. At the start of the game, you can set the speed of the zombies.

• If you are ever in the same room as a zombie, or if two zombies are on either side of you in a hallway, you get eaten (and lose the game).

Defining Zombie Graphs

• You know where all the zombies are at all times.

• The zombie number of a graph is the minimum number of zombies needed to eventually catch and eat you assuming you use the best possible strategy.

Examples

• A path has zombie number 1.

Examples

• A tree has zombie number 1.

Examples

• A cycle has zombie number 2.

• Thus, a graph has zombie number 1 if and only if it is a tree.

Examples

has zombie number 3. If only 2 zombies are on , you can always escape by moving towards an unoccupied vertex.

Examples

has zombie number 3. If 3 zombies are on , you will be eaten.

• In general, has zombie number .

Cops and Robbers

There is a similar well-known game:• A robber runs around a graph trying to escape

cops, who travel by helicopter between adjacent vertices.

The difference between the two games: • Zombies travel on edges. • Cops do not travel on edges. Instead they

travel between adjacent vertices.

Cops and Robbers

The zombie can catch the person:

The cop cannot catch the robber:

Cop Number

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 .

Theorem. The zombie number of a graph is either or .

The following graph has cop number 3 and zombie number 2:

Theorem. The zombie number of a graph is either or .

The following graph has cop number 3 and zombie number 3.

If there are only 2 zombies, you can always move to whichever of the three vertices is the furthest from both zombies.

Theorem. The zombie number of a graph is either or .

A graph with cop number 3:

Theorem. The zombie number of a graph is either or .

3 zombies can catch you on this graph.

Theorem. The zombie number of a graph is either or .

3 zombies can catch you on this graph.

Theorem. The zombie number of a graph is either or .

3 zombies can catch you on this graph.

Theorem. The zombie number of a graph is either or .

3 zombies can catch you on this graph.

Forbidden Minors for Zombie number 2

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.

Further Questions

• Which graphs have zombie number 3?

• Zombie number 4? 5? 6?

• If the cop number of the graph is known, how hard is it to determine the zombie number?

The End