In Thomassen's celebrated proof, he refers to maximal segments of a graph. My first question is would someone know what he means?
Specifically in the proof of Lemma 2.10, where he mentions "at least two disjoint maximal segments"
He also declares $P'$ to be a shortest path between two subgraphs, I think, namely $P$ and $C-V(P)$. My second question is isn't a shortest path just a single edge, namely one that in $C$ connects $P$ with the rest of $C$? He using the word shortest path made me think he is saying I might need more than one edge. I would appreciate any drawing or picture of this proof.