### 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(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 - 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) |
---|---|

Pages (from-to) | 1322-1335 |

Number of pages | 14 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2010 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*24*(4), 1322-1335. https://doi.org/10.1137/100791130

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

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 24, no. 4, pp. 1322-1335. https://doi.org/10.1137/100791130

}

TY - JOUR

T1 - Surviving rates of graphs with bounded treewidth for the firefighter problem

AU - Cai, Leizhen

AU - Cheng, Yongxi

AU - Verbin, Elad

AU - Zhou, Yuan

PY - 2010/12/1

Y1 - 2010/12/1

N2 - 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.

AB - 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.

KW - Firefighter problem

KW - Outerplanar graph

KW - Surviving rate

KW - Tree

KW - Treewidth

UR - http://www.scopus.com/inward/record.url?scp=79251579634&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251579634&partnerID=8YFLogxK

U2 - 10.1137/100791130

DO - 10.1137/100791130

M3 - Article

AN - SCOPUS:79251579634

VL - 24

SP - 1322

EP - 1335

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -