Shattering-extremal set systems of VC dimension at most 2

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=\{F \cap S : F \in \mathcal{F}\}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question from \cite{VC1} about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.