Exactly Hittable Interval Graphs

TL;DR

This work studies exactly hittable interval graphs (EHIG), where an interval model must be exactly hit by a hitting set. It introduces a canonical interval representation $H_G$ built from a maximal-clique order and gadgetry, enabling a structural analysis of exact hittability. A forbidden-structure characterization is proved: an interval graph $G$ is EHIG if and only if it avoids all graphs in the forbidden family $\mathcal{F}$ as induced subgraphs, and this leads to a polynomial-time recognition approach via reduction to the MMSC problem on $H_G$. The results place Proper Interval Graphs as a strict subset of EHIG, establish a practical recognition method, and provide a foundation for further explorations of exact-hit representations in broader graph classes.

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.

Figures (6)

• Figure 3: Construction of Canonical Interval Representation (i) Interval Graph $G$ with its maximal cliques $Q_1,Q_2,Q_3,Q_4$(ii) Linear ordering of maximal cliques $\mathcal{O} = \{Q_1,Q_2,Q_3,Q_4\}$(iii) Interval representation of $G$ obtained from $\mathcal{O}$(iv) Gadgets $D_1$ to $D_4$(v) Canonical interval representation for $G$
• Figure 4: Stretching intervals to the left
• Figure 5: Construction of path $P$
• Figure 6: Forbidden structure formation
• Figure 7: Vertices $v_i$ and $v_j$ belong to three consecutive cliques in the clique cover

Theorems & Definitions (29)

• Lemma 1
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• Lemma 2
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• Lemma 4
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