Lower and Upper Bounds for Long Induced Paths in 3-Connected Planar Graphs

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) is an element of Omega((log(2) n)(1-epsilon)), where epsilon is any positive constant smaller than 1, and describe an infinite family of 3-connected planar graphs for which p(G) is an element of 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 root n vertices. The proofs in the paper are constructive and give rise to O(n)-time algorithms.