Contract grant sponsor: ANR Project HEREDIA; Contract grant number: ANR-10-JCJC-HEREDIA; Contract grant sponsor: CNRS (to JvdH).
Fire Containment in Planar Graphs
Article first published online: 3 AUG 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 73, Issue 3, pages 267–279, July 2013
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
- Issue published online: 10 MAY 2013
- Article first published online: 3 AUG 2012
- Manuscript Revised: 15 MAY 2012
- Manuscript Received: 11 MAR 2011
- ANR Project HEREDIA. Grant Number: ANR-10-JCJC-HEREDIA
- the Firefighter Problem;
- surviving rate;
- planar graphs
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.