  Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where
$\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of
$\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$
such that each subset in $\mathcal{S}$ contains exactly one element in
$\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact
hitting set. In this paper, we study interval graphs which have intersection
models that are exactly hittable. We refer to these interval graphs as exactly
hittable interval graphs (EHIG). We present a forbidden structure
characterization for EHIG. We also show that the class of proper interval
graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in
polynomial time to recognize graphs belonging to the class of EHIG.
