Contract grant sponsor: ANR Project HEREDIA; Contract grant number: ANR-10-JCJC-HEREDIA; Contract grant sponsor: CNRS (to JvdH).
1097-0118/asset/JGT_left.gif?v=1&s=1444a0be416c8f041967dd96201bc2e34f807002)
Article
Fire Containment in Planar Graphs
Article first published online: 3 AUG 2012
DOI: 10.1002/jgt.21673
© 2012 Wiley Periodicals, Inc.
Additional Information
How to Cite
Esperet, L., van den Heuvel, J., Maffray, F. and Sipma, F. (2013), Fire Containment in Planar Graphs. J. Graph Theory, 73: 267–279. doi: 10.1002/jgt.21673
Publication History
- Issue published online: 10 MAY 2013
- Article first published online: 3 AUG 2012
- Manuscript Revised: 15 MAY 2012
- Manuscript Received: 11 MAR 2011
Funded by
- ANR Project HEREDIA. Grant Number: ANR-10-JCJC-HEREDIA
- CNRS
- Abstract
- Article
- References
- Cited By
Keywords:
- the Firefighter Problem;
- surviving rate;
- planar graphs
Abstract
In a graph G, a fire starts at some vertex. At every time step, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. The k-surviving rate 
of G is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs 
, we are interested in the minimum value k such that for some constant 
and all 
, 
(i.e., such that linearly many vertices are expected to be saved in every graph from 
). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.