Exactly Hittable Interval Graphs

TL;DR

This paper characterizes exactly hittable interval graphs, introduces a forbidden structure characterization, shows proper interval graphs are a strict subset, and provides a polynomial-time recognition algorithm.

Contribution

It introduces the class of exactly hittable interval graphs, characterizes them via forbidden structures, and offers an efficient recognition algorithm.

Findings

Forbidden structure characterization for EHIG

Proper interval graphs are a strict subclass of EHIG

Polynomial-time recognition algorithm for EHIG

Abstract

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.