Approximation Algorithms for the Firefighter Problem: Cuts over Time and Submodularity
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
The surviving rate of an infected network
Theoretical Computer Science
The surviving rate of an outerplanar graph for the firefighter problem
Theoretical Computer Science
Surviving Rates of Graphs with Bounded Treewidth for the Firefighter Problem
SIAM Journal on Discrete Mathematics
The surviving rate of planar graphs
Theoretical Computer Science
Fighting constrained fires in graphs
Theoretical Computer Science
The 2-surviving rate of planar graphs without 4-cycles
Theoretical Computer Science
Discrete Applied Mathematics
The 2-surviving rate of planar graphs without 6-cycles
Theoretical Computer Science
A lower bound of the surviving rate of a planar graph with girth at least seven
Journal of Combinatorial Optimization
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We consider the following firefighter problem on a graph $G=(V,E)$. Initially, a fire breaks out at a vertex $v$ of $G$. In each subsequent time unit, a firefighter protects one vertex, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. Let $\mathrm{sn}(v)$ denote the maximum number of vertices the firefighter can save when a fire breaks out at vertex $v$ of $G$. We define the surviving rate $\rho(G)$ of $G$ to be the average percentage of vertices that can be saved when a fire randomly breaks out at a vertex of $G$, i.e., $\rho(G)=\sum_{v\in V}\mathrm{sn}(v)/n^2$. In this paper, we prove that for every tree $T$ on $n$ vertices, $\rho(T)1-\sqrt{2/n}$. Furthermore, we show that $\rho(G)1/6$ for every outerplanar graph $G$, and $\rho(H)3/10$ for every Halin graph $H$ with at least 5 vertices.