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Outerplanar obstructions for the feedback vertex set. (English) Zbl 1273.05213
Nešetřil, Jaroslav (ed.) et al., Extended abstracts of the 5th European conference on combinatorics, graph theory and applications, EuroComb’09, Bordeaux, France, September 7–11, 2009. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 34, 167-171 (2009).
Summary: For \(k\geq 1\), let \({\mathcal{F}}_{k}\) be the class containing every graph that contains \(k\) vertices meeting all its cycles. The minor-obstruction set for \({\mathcal{F}}_{k}\) is the set \(\mathsf{obs}(F_{k})\) containing all minor-minimal graph that does not belong to \({\mathcal{F}}_{k}\). We denote by \({\mathcal{Y}}_{k}\) the set of all outerplanar graphs in \(\mathsf{obs}({\mathcal{F}}_{k})\). In this paper, we provide a precise characterization of the class \({\mathcal{Y}}_{k}\). Then, using the symbolic method, we prove that \(|{\mathcal{Y}}_{k}| \sim \alpha \cdot k^{-5/2} \cdot \rho ^{-k}\) where \(\alpha \doteq 0.02602193\) and \(\rho^{-1} \doteq 14.49381704\).
For the entire collection see [Zbl 1239.05008].

05C83 Graph minors
05C30 Enumeration in graph theory
Full Text: DOI
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