Making Your House Safe From Zombie Attacks - PowerPoint PPT Presentation

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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. ...

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Making Your House Safe From Zombie Attacks

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Making your house safe from zombie attacks l.jpg

Making Your House Safe From Zombie Attacks

Jim Belk and Maria Belk


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How can we construct a house so that we

can escape from  grizzly bears?

Let’s make this more precise.


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Defining Grizzly Bear Graphs

  • We represent the house by a graph.


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Defining Grizzly Bear Graphs

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


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Defining Grizzly Bear Graphs

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


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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.


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Defining Grizzly Bear Graphs

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


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Defining Zombie Graphs

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


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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).


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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.


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Examples

  • A path has zombie number 1.


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Examples

  • A tree has zombie number 1.


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Examples

  • A cycle has zombie number 2.

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


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Examples

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


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Examples

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

  • In general,  has zombie number .


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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.


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Cops and Robbers

The zombie can catch the person:

The cop cannot catch the robber:


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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 .


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Theorem. The zombie number of a graph  is either  or .

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


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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.


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Theorem. The zombie number of a graph  is either  or .

A graph with cop number 3:


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Theorem. The zombie number of a graph  is either  or .

3 zombies can catch you on this graph.


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Theorem. The zombie number of a graph  is either  or .

3 zombies can catch you on this graph.


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Theorem. The zombie number of a graph  is either  or .

3 zombies can catch you on this graph.


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Theorem. The zombie number of a graph  is either  or .

3 zombies can catch you on this graph.


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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.


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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?


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The End


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