Lovasz's Path removal conjecture

Question

The Lovász Path Removal Conjecture states:

For any positive integer $k$, there exists a minimum positive integer $f(k)$ such that, for any two vertices $x$, $y$ in any $f(k)$-vertex-connected graph $G$, there is an $x$-$y$ path $P$ in $G$ such that $G\backslash P$ is $k$-vertex-connected.

It is commonly stated that $ f(1)=3 $ due to a theorem in Tutte's paper "How to draw a graph" (Proc. London Math. Soc. 13.3 (1963), 743–768).

However, I can't find any theorem in Tutte's paper that directly states that a 3-vertex-connected graph contains a non-separating path between any two vertices. Can someone point out to me the relevant theorem in Tutte's paper and show how it implies the above fact?

Answer 1

Tutte in that paper shows that if the connectivity of a graph is at least 3, then for any two distinct vertices in that graph, there exists a non-separating path between them. Any cycle plus a single chord shows $f(1) \geq 3$ and Tutte's Wheel Theorem shows that $f(1) \leq3$. Thus, $f(1)=3$.