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.

翻译：设集合系统 $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$，其中 $\mathcal{U}$ 是元素集，$\mathcal{S}$ 是 $\mathcal{U}$ 的子集族。精确击中集 $\mathcal{U}'$ 是 $\mathcal{U}$ 的子集，使得 $\mathcal{S}$ 中的每个子集恰好包含 $\mathcal{U}'$ 中的一个元素。若一个集合系统存在精确击中集，则称其为精确可击中的。本文研究具有精确可击中交模型的区间图，并将此类区间图称为精确可击中区间图（EHIG）。我们给出了 EHIG 的禁止结构刻画，同时证明了真区间图类是 EHIG 的一个严格子类。最后，我们提出了一种多项式时间算法，用于识别属于 EHIG 类的图。