Making Your House Safe From Zombie Attacks

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## Making Your House Safe From Zombie Attacks

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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?