Statements related to Thurston's work on the surface

Question

If we have simple closed curves $\alpha$ and $\beta$ on a surface $\Sigma_g$, the intersection number $i(\alpha ,\beta)$ is defined to be the minimal cardinality of $\alpha_1\cap\beta_1$ as $\alpha_1$ and $\beta_1$ range over all simple closed curves isotopic to $\alpha$ and $\beta$, respectively. We say $\alpha$ and $\beta$ intersect minimally if $i(\alpha ,\beta) = |\alpha\cap\beta|\,$.

How to see that $\alpha$ and $\beta$ intersect minimally if there are no pairs of $p,q\in\alpha\cap\beta$ such that the arc joining $p$ to $q$ along $\alpha$ followed by the arc from $q$ back to $p$ along $\beta$ bounds a disk in $\Sigma_g$?
Maybe a sketch of the proof idea?

I think the converse is also true : "that $\alpha$ and $\beta$ intersect minimally only if there are no pairs of $p,q\in\alpha\cap\beta$ such that the arc joining $p$ to $q$ along $\alpha$ followed by the arc from $q$ back to $p$ along $\beta$ bounds a disk in $\Sigma_g$."

Answer 1

This is called the "bigon criterion". For a discussion, see Section 1.2.4 (and in particular Proposition 1.7) of the "Primer on mapping class groups" by Farb and Margalit.

The Google search "bigon criterion" also finds various references and lecture notes. For example, here is the top hit:

https://math.stackexchange.com/questions/1646340/proof-of-the-bigon-criterion