Making Your House Safe From Zombie Attacks

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# Making Your House Safe From Zombie Attacks - PowerPoint PPT Presentation

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.

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### 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 .

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.

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?