Lower and upper bounds for long induced paths in 3-connected planar graphs

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) ≥ 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(log n). 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 3√n vertices. The proofs in the paper are constructive and give rise to O(n)-time algorithms. © 2013 Springer-Verlag.