Tight bounds on the Carathéodory and exchange numbers in $\triangle$-convexity

The $\triangle$-convexity space on a finite, simple graph $G = (V,E)$ is the collection $\mathcal{C}$ of subsets $S \subseteq V(G)$ such that whenever $x \in V(G)$ forms a triangle with two vertices in $S$, we have $x \in S$. The members of $\mathcal{C}$ are called convex sets, and the convex hull o