On a pursuit game on Cayley graphs
Combinatorica
Explicit construction of linear sized tolerant networks
Discrete Mathematics - First Japan Conference on Graph Theory and Applications
Fire containment in grids of dimension three and higher
Discrete Applied Mathematics
The surviving rate of an infected network
Theoretical Computer Science
The Surviving Rate of a Graph for the Firefighter Problem
SIAM Journal on Discrete Mathematics
Surviving Rates of Graphs with Bounded Treewidth for the Firefighter Problem
SIAM Journal on Discrete Mathematics
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The firefighter problem is a simplified model for the spread of a fire (or disease or computer virus) in a network. A fire breaks out at a vertex in a connected graph, and spreads to each of its unprotected neighbours over discrete time-steps. A firefighter protects one vertex in each round which is not yet burned. While maximizing the number of saved vertices usually requires a strategy on the part of the firefighter, the fire itself spreads without any strategy. We consider a variant of the problem where the fire is constrained by spreading to a fixed number of vertices in each round. In the two-player game of k-firefighter, for a fixed positive integer k, the fire chooses to burn at most k unprotected neighbours in a given round. The k-surviving rate of a graph G is defined as the expected percentage of vertices that can be saved in k-firefighter when a fire breaks out at a random vertex of G. We supply bounds on the k-surviving rate, and determine its value for families of graphs including wheels and prisms. We show using spectral techniques that random d regular graphs have k-surviving rate at most (1+O(d^-^1^/^2))k+1. We consider the limiting surviving rate for countably infinite graphs. In particular, we show that the limiting surviving rate of the infinite random graph can be any real number in [1/(k+1),1].