Abstract
Let G be a 3-connected planar graph with n vertices and let p(G) be the maximum number of vertices of an induced subgraph of G that is a path. We prove that \(p(G) \geq \frac{\log n}{12 \log \log n}\). To demonstrate the tightness of this bound, we notice that the above inequality implies p(G) ∈ Ω((log2 n)1 − ε), where ε is any positive constant smaller than 1, and describe an infinite family of 3-connected planar graphs for which p(G) ∈ O(logn). As a byproduct of our research, we prove a result of independent interest: Every 3-connected planar graph with n vertices contains an induced subgraph that is outerplanar and connected and that contains at least \(\sqrt[3] n\) vertices. The proofs in the paper are constructive and give rise to O(n)-time algorithms.
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Di Giacomo, E., Liotta, G., Mchedlidze, T. (2013). Lower and Upper Bounds for Long Induced Paths in 3-Connected Planar Graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_19
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DOI: https://doi.org/10.1007/978-3-642-45043-3_19
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