The firefighter problem for graphs of maximum degree three

We consider a dynamic problem introduced by B. Hartnell in 1995 [4]. Let (G, r) be a connected rooted graph (where r ∈ V (G)). At time 0, a fire breaks out at r. At each subsequent time interval, the firefighter defends some vertex which is not on fire, and then the fire spreads to all undefended neighbours of each burning (i.e., on fire) vertex. Once defended, a vertex remains so for all time intervals. The process ends when the fire can no longer spread. The firefighter (optimization) problem is to determine the maximum number of vertices that can be saved, i.e., that are not burning when the process ends. Papers investigating the firefighter problem have appeared in the literature. Algorithms for twoand three-dimensional grid graphs are presented in [9]. These lead to bounds on the maximum number of vertices that can be saved. NP-completeness of the firefighter problem on bipartite graphs is ∗Research supported by The Natural Sciences and Engineering Council of Canada.