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Planar graphs without chordal 5-cycles are 2-good. (English) Zbl 1393.05094
Summary: Let \(G\) be a connected graph with \(n\geq 2\) vertices. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects two vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let \(\mathrm{sn}_2(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The 2-surviving rate \(\rho_2(G)\) of \(G\) is defined to be the real number \(\frac{1}{n^2} \sum_{v\in V(G)} \mathrm{sn}_2(v)\). Then it is obvious that \(0<\rho_2(G)<1\). The graph \(G\) is called 2-good if there is a constant \(c>0\) such that \(\rho_2(G)>c\). In this paper, we prove that every planar graph with \(n\geq 2\) vertices and without chordal 5-cycles is 2-good.

05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
05C40 Connectivity
Full Text: DOI
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