# Surviving Rates of Graphs with Bounded Treewidth for the Firefighter Problem

## Abstract

*surviving rate*of a graph is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of the graph, which measures the defense ability of a graph as a whole. In this paper, we study the surviving rates of graphs with bounded treewidth. We prove that the surviving rate of every

*n*-vertex outerplanar graph is at least $1-\Theta(\frac{\log n}{n})$, which is asymptotically tight. We also prove that if

*k*firefighters are available in each round, then the surviving rate of an

*n*-vertex graph with treewidth at most

*k*is at least $1-O(\frac{k^{2}\log n}{n})$. Furthermore, we show that the greedy strategy of Hartnell and Li [

*Congr. Numer.*, 145 (2000), pp. 187–192] for trees saves at least $1-\Theta(\frac{\log n}{n})$ percent of vertices on average for an

*n*-vertex tree. Our results settle a conjecture and two problems of Cai and Wang [

*SIAM J. Discrete Math.*, 23 (2009), pp. 1814–1826] in affirmative.

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## References

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#### History

**Submitted**: 5 April 2010

**Accepted**: 29 July 2010

**Published online**: 12 October 2010

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