Non-separating paths in 4-connected graphs

In 1975, Lovasz conjectured that for any positive integer k, there exists a minimum positive integer f (k) such that, for any two vertices x, y in any f (k)-connected graph G, there is a path P from x to y in G such that G V ( P) is k-connected. A result of Tutte implies f ( 1) = 3. Recently, f ( 2) = 5 was shown by Chen et al. and, independently, by Kriesell. In this paper, we show that f ( 2) = 4 except for double wheels.