3-connected multigraph and parallel edge

Question

In Diestel book, 5th edition, page 62, he says in the footnote "Recall also that 3-connected multigraphs cannot have multiple edges". Also, in the proof for Lemma 3.2.1, he uses "the fact" that "$G\,\dot{-}\,e$ graph has no parallel edges."

Isn't it obviously false? I can start with any 3-connected graph and start adding arbitrary edges and parallel edges and it will still remain 3-connected. What am I missing here?

Answer 1

As is often the case, the problem is cleared up when we see the part of the context that you didn't tell us about. Immediately before the sentence that you quote, in that same footnote, he suggests to “See Chapter $1.10$ for the formal definition of suppressing vertices in a multigraph”. Chapter $1.10$ contains the following explanation:

A few differences, however, should be pointed out. [...] The ends of loops and parallel edges in a multigraph $G$ are considered as separating that edge from the rest of $G$. [...] Thus, a multigraph with a loop is never $2$-connected, and any $3$-connected multigraph is in fact a graph.