In Farb and Margalit's A Primer on Mapping Class Groups, the proof of Theorem 3.10 uses the notion of graph automorphisms. I don't know much about graph theory and have trouble following their argument.
Consider the following graph $\Gamma$:
[...], the only automorphisms of $\Gamma$ come from flipping the three edges that form loops and swapping pairs of edges that form a loop. In particular, any automorphism of $\Gamma$ must fix the three edges coming from $\alpha_4$. Thus, we see that $\phi$ preserves the orientation of $\alpha_4$, [...]
The edges of this graph come from simple closed curves on a surface $S$ and the vertices are the intersections. $\alpha_4$ is the curve represented by the loop containing three edges and $\phi$ is an orientation-preserving homeomorphism on $S$ fixing $\Gamma$.
Questions 1: What is meant by flipping the three edges? Permuting them? Rotating them? Something else entirely?
Question 2: How can I convince myself that those automorphisms are the only automorphisms?
Question 3: Usually in the book, if a map $f$ fixes a set $M$, usually $f(M) \subseteq M$ is meant. Do Farb and Margalit mean that each edge of $\alpha_4$ is fixed individually here? That would make more sense in the context, but not coincide with the usual use of this notion in the book.
