RTD-Conjecture and Concept Classes Induced by Graphs

It is conjectured that the recursive teaching dimension of any finite concept class is upper-bounded by the VC-dimension of this class times a universal constant. In this paper, we confirm this conjecture for two rich families of concept classes where each class is induced by some graph $G$. For each $G$, we consider the class whose concepts represent stars in $G$ as well as the class whose concepts represent connected sets in $G$. We show that, for concept classes of this kind, the recursive teaching dimension either equals the VC-dimension or is less by $1$.