# Surviving rates of graphs with bounded treewidth for the firefighter problem

Leizhen Cai, Yongxi Cheng, Elad Verbin, Yuan Zhou

Research output: Contribution to journalArticle

### Abstract

The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, 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. The 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 - Θ(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(k2/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 - Θ(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.

Original language English (US) 1322-1335 14 SIAM Journal on Discrete Mathematics 24 4 https://doi.org/10.1137/100791130 Published - Dec 1 2010 Yes

### Fingerprint

Bounded Treewidth
Graph in graph theory
Vertex of a graph
Outerplanar Graph
Treewidth
Percent
Percentage
Discrete-time
Game

### Keywords

• Firefighter problem
• Outerplanar graph
• Surviving rate
• Tree
• Treewidth

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

Surviving rates of graphs with bounded treewidth for the firefighter problem. / Cai, Leizhen; Cheng, Yongxi; Verbin, Elad; Zhou, Yuan.

In: SIAM Journal on Discrete Mathematics, Vol. 24, No. 4, 01.12.2010, p. 1322-1335.

Research output: Contribution to journalArticle

Cai, Leizhen ; Cheng, Yongxi ; Verbin, Elad ; Zhou, Yuan. / Surviving rates of graphs with bounded treewidth for the firefighter problem. In: SIAM Journal on Discrete Mathematics. 2010 ; Vol. 24, No. 4. pp. 1322-1335.
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