TL;DR
This paper establishes tight bounds on the number of firefighters needed to contain a fire in planar grid graphs, resolving two conjectures and advancing understanding of epidemic control models.
Contribution
It provides the first tight bounds for firefighter numbers in Cartesian and strong planar grids, confirming two prior conjectures.
Findings
Tight bounds on firefighters needed for Cartesian planar grids.
Tight bounds on firefighters needed for strong planar grids.
Resolution of two conjectures by Ng and Raff.
Abstract
The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.
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