Assume that a fire spreads in an grid where some fire fighters are deployed. Initially, all cells of the grid are on fire, except those occupied by a fighter. In each step, fighters can move to adjacent cells and put their fire out, while each burning cell ignites all neighbors not protected by a fighter. The question, also known as a lion-and-man problem [3], is how many fighters are needed to completely extinguish the fire. A column of n fighters can sweep the grid and erase the fire. Whether fighters are sufficient is still an open problem.
This note presents the following structural property that holds for fire-fighting in arbitrary undirected graphs, including grid graphs as a special case.
Suppose the fighters perform a sequence of moves, M, that transform configuration A into configuration B. If we swap the states of all vertices in B that do not contain a fighter and let the fighters run backwards, we reach a configuration where each vertex burning in A is now free of fire. As a consequence, if M is an extinguishing strategy, so is its reverse, .
