Plane graphs of diameter two are 2-optimal

Jiangxu Kong, Yiqiao Wang, Jiacheng Hu, Yang Wang and Weifan Wang

Applied Mathematics and Computation, 2023, vol. 441, issue C

Abstract: Let G be a connected graph with n vertices. Suppose a fire breaks out at some vertex of G. At each time interval, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. Let dnk(v) denote the minimum number of vertices that the fire may damage when a fire breaks out at vertex v. The k-expected damage of G, denoted by εk(G), is the expectation of the proportion of vertices that can be damaged from the fire, if the starting vertex of the fire is chosen uniformly at random, i.e., εk(G)=∑v∈V(G)dnk(v)/n2. A class of graphs G is called k-optimal if εk(G) tends to 0 as n tends to infinity for any G∈G. In this paper, we prove that planar graphs of diameter two are 2-optimal, which is the best possible.

Keywords: Firefighter Problem; Plane graph; Diameter two (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:441:y:2023:i:c:s0096300322007858

DOI: 10.1016/j.amc.2022.127717

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