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Continuous and discrete models for firefighting problems are well-studied in Theoretical Computer Science. We introduce a new, discrete, and more general framework based on a hexagonal cell graph to study firefighting problems in varied terrains. We present three different firefighting problems in the context of this model; for two of which, we provide efficient polynomial time algorithms and for the third, we show NP-completeness. We also discuss possible extensions of the model and their implications on the computational complexity., Comment: 25 pages, 13 figures, pre-print of an article presented at CALDAM 2020

In 2006, Alberto Bressan suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time $t$ the total length of all barriers built so far must not exceed $t \cdot v$, where $v$ is a speed constant. How large a speed $v$ is needed? He proved that speed $v>2$ is sufficient, and that $v>1$ is necessary. This gap of $(1,2]$ is still open. The crucial question seems to be the following. {\em When trying to contain a fire, should one build, at maximum speed, the enclosing barrier, or does it make sense to spend some time on placing extra delaying barriers in the fire's way?} We study the situation where the fire must be contained in the upper $L_1$ half-plane by an infinite horizontal barrier to which vertical line segments may be attached as delaying barriers. Surprisingly, such delaying barriers are helpful when properly placed. We prove that speed $v=1.8772$ is sufficient, while $v >1.66$ is necessary., Comment: 15 pages, 10 figures, pre-print of an article published in WADS 2019