We show that the firefighter problem is NP-complete for cubic graphs. We also show that given a rooted tree of maximum degree three in which every leaf is the same distance from the root, it is NPcomplete to decide whether or not there is a strategy that protects every leaf from the fire, which starts at the root. By contrast, we describe a polynomial time algorithm to decide if it is possible to save a given subset of the vertices of a graph with maximum degree three, provided the fire breaks out at a vertex of degree at most two.
REF showed that for a tree of degree at most 3, it is NP-hard to save a critical set with budget of one firefighter per round, but is polynomial time when the fire starts from a vertex of degree at most 2.
The Firefighter Problem For Cubic Graphs
Discret. Math.
Discret. Math.
Computer Science Mathematics