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 unburnt
neighbourhood of all the burning vertices and the firefighters again protect
b vertices. Once a vertex has been burnt or protected it remains that way for
the rest of the game. We previously introduced the concept of
distance-restricted firefighting where 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 the
Maximum Vertices Saved problem and present an integer program for
solving these problems. In the penultimate section we also discuss some
interesting properties of the Expected Damage function