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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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Presentation Transcript
slide5
How can we construct a house so that we

can escape from  grizzly bears?

Let’s make this more precise.

defining grizzly bear graphs
Defining Grizzly Bear Graphs
  • We represent the house by a graph.
defining grizzly bear graphs7
Defining Grizzly Bear Graphs
  • We represent the house by a graph. Vertices represent rooms.
defining grizzly bear graphs8
Defining Grizzly Bear Graphs
  • We represent the house by a graph. Vertices represent rooms, and edges represent hallways.
defining grizzly bear graphs9
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 graphs10
Defining Grizzly Bear Graphs
  • You move much, much faster than the grizzly bears.
defining zombie graphs
Defining Zombie Graphs
  • You move much, much faster than the grizzly bears zombies.
defining zombie graphs12
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 graphs13
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
Examples
  • A path has zombie number 1.
examples15
Examples
  • A tree has zombie number 1.
examples16
Examples
  • A cycle has zombie number 2.
  • Thus, a graph has zombie number 1 if and only if it is a tree.
examples17
Examples
  •  has zombie number 3. If only 2 zombies are on , you can always escape by moving towards an unoccupied vertex.
examples18
Examples
  •  has zombie number 3. If 3 zombies are on , you will be eaten.
  • In general,  has zombie number .
cops and robbers
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 robbers20
Cops and Robbers

The zombie can catch the person:

The cop cannot catch the robber:

cop number
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
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 or23
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 or24
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 or25
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 or26
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 or27
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 or28
Theorem. The zombie number of a graph  is either  or .

3 zombies can catch you on this graph.

forbidden minors for zombie number 2
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
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?