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：给定一个集合系统 $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$，其中 $\mathcal{U}$ 是元素集合，$\mathcal{S}$ 是 $\mathcal{U}$ 的子集族，精确击中集 $\mathcal{U}'$ 是 $\mathcal{U}$ 的一个子集，使得 $\mathcal{S}$ 中的每个子集恰好包含 $\mathcal{U}'$ 中的一个元素。如果一个集合系统存在精确击中集，则称其为精确可击的。本文研究具有精确可击交模型的区间图，将这些区间图称为精确可击区间图（EHIG）。我们给出了EHIG的禁止结构刻画，并证明了真区间图类是EHIG的真子类。最后，我们提出了一个多项式时间算法来识别属于EHIG类的图。