Fighting the Plane Patricia Fogarty University of Vermont January 17, 2003

Abstract Given any type of grid and k firefighters, suppose that a fire breaks out at one of the vertices. At the next time interval, we protect k vertices, and then the fire spreads to any unprotected vertices. We want to find patterns with these grids, and from these, we can determine the minimum number of vertices burned in rectangular grids and in the plane.

Main Idea How does it work? A fire starts at some vertex in the grid

Then k firefighters protect vertices in the grid let k = 1

Then the fire spreads to all adjacent vertices

Objective build walls that will allow us to protect the whole grid protect the grid so that a minimum number of vertices burn

Rectangular Grids One firefighter Restrict the y-coordinate to go from 0 to n Let the x-coordinate extend to infinity Here n = 10

Let’s start at the top What does this look like? How can we protect the grid First pattern: protect one side at a time Once one side is completely protected we go to the second

Second Pattern Protect the left side of the grid Protect the right side of the grid Alternate between these two sides

Start somewhere else Does the first pattern hold?

Start somewhere else Does the second pattern hold?

The Plane Now we are unrestricted in the y- coordinate i.e. The Cartesian Plane Can we contain the fire with one firefighter?

The Plane Can we contain the fire with two firefighters? How long does it take?

The Plane Pattern Protect the bottom portion Protect the upper portion Takes 8 time intervals

Protecting the Plane Later What happens when we do not protect the grid for x time intervals? Can we protect the grid with two firefighters? look at x = 1

x = 1 and conclusions How long does it take? 32x+1 How many vertices are burned? 318x 2 +14x+1

Connections Virus Control on Networks Wildfires Dennis E. Shasha. Wildfires. Dr. Dobb’s Journal, January 2001, pages

More Research The quarter plane More than one fire Hexagonal grids 3D grids

Bibliography Ping Wang and Stephanie A. Moeller. Fire Control on Graphs. Journal of Combinatorial Mathematics and Combinatorial Computing 41, pages 19-34, A.S. Finbow and B.L. Hartnell. On designing a network to defend against random attacks on radius two. Networks, 19: , G. Gunther and B.L. Hartnell. Graphs with isomorphic survivor graphs. Congressus Numerantium 79, pages , B.L. Hartnell. The optimum defense against random subersions in a network. Congressus Numerantium 24, pages , 1979.

Contact Information Web page: Follow links to fire control on graphs applet