3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.Comment: 8 page

Orienting edges to fight fire in graphs

International audienceWe investigate a new oriented variant of the Firefighter Problem. In the traditional Firefighter Problem, a fire breaks out at a given vertex of a graph, and at each time interval spreads to neighbouring vertices that have not been protected, while a constant number of vertices are protected at each time interval. In the version of the problem considered here, the firefighters are able to orient the edges of the graph before the fire breaks out, but the fire could start at any vertex. We consider this problem when played on a graph in one of several graph classes, and give upper and lower bounds on the number of vertices that can be saved. In particular, when one firefighter is available at each time interval, and the given graph is a complete graph, or a complete bipartite graph, we present firefighting strategies that are provably optimal. We also provide lower bounds on the number of vertices that can be saved as a function of the chromatic number, of the maximum degree, and of the treewidth of a graph. For a subcubic graph, we show that the firefighters can save all but two vertices, and this is best possible

Cops, robbers and firefighters on graphs

This thesis focuses on the game of cops and robbers on graphs, which was introduced independently by Quilliot in 1978 and by Nowakowski and Winkler in 1983, and one of its variants, the firefighter problem. In the game of cops and robbers, the cops start by choosing their starting positions on vertices of a graph, then the robber chooses his starting point. Then, they move each in turn along the edges of the graph. The basic objective is to determine whether the cops have a strategy which allows them to catch the robber. Looped vertices allow the cops and the robber to pass their turn. The first chapter explores the effect of loops on the cop number and the capture time. It provides examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play. In the second chapter, we investigate the links between this game and algebraic topology. We extend the game of cops and robbers on graphs by considering the case where the cops chase the image of the robber by a graph homomorphism. We prove that the cop number associated with a graph homomorphism is a homotopic invariant. Homotopies between graph homomorphisms or homotopy equivalences between graphs allow us to compare their cop numbers and also their capture times. Finally, using homotopic invariants such as homology groups, we investigate structural properties of copwin graphs. Finally, in the third chapter, we explore the Firefighter problem, introduced by Hartnell in 1995, where a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. While focusing on the case of trees, we also consider a variant game called Fractional Firefighter in which the amount of protection allocated to a vertex lies between 0 and 1. While most of the work in this area deals with a constant amount of firefighters available at each turn, we consider three research questions which arise when including the sequence of firefighters as part of the instance. We first introduce an online version of both Firefighter and Fractional Firefighter, in which the number of firefighters available at each turn is revealed over time. We show that a greedy algorithm on finite trees is 1/2-competitive for both online versions, which generalises a result previously known for special cases of Firefighter. We also show that the optimal competitive ratio of online Firefighter ranges between 1/2 and the inverse of the golden ratio. Next, given two firefighter sequences, we discuss sufficient conditions for the existence of an infinite tree that separates them, in the sense that the fire can be contained with one sequence but not with the other. To this aim, we study a new purely numerical game called targeting game. Finally, we give sufficient conditions for the fire to be contained on infinite trees, expressed as the asymptotic comparison of the number of firefighters and the size of the tree levels