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
The following graph has cop number 3 and zombie
to whichever of the three vertices is the furthest
number 3.
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