Distance-Restricted Firefighting on Finite Graphs

In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph $G$ and then $b$ firefighters protect $b$ vertices. On each subsequent turn, the fire spreads to the collective unburned neighbourhood of all the burning vertices and the firefighters again protect $b$ vertices. Once a vertex has been burned or protected it remains that way for the rest of the game. In $\mathit{\text{distance-restricted}}\mathit{\text{firefighting}}$ the firefighters’ movement is restricted so they can only move up to some fixed distance $d$ and they may or may not be permitted to move through burning vertices. In this paper we establish the NP-completeness of the distance-restricted versions of {\sc $b$-Firefighter} and present an integer program for computing the exact value. We also discuss some interesting properties of the $\mathit{\text{Expected}}\mathit{\text{Damage}}$ function.