A characterization of interval nest digraphs

TL;DR

This paper provides a complete characterization of interval nest digraphs using vertex orderings with forbidden patterns, advancing the understanding of their structure within the broader class of interval digraphs.

Contribution

It introduces a novel vertex-ordering characterization of interval nest digraphs through forbidden patterns, completing the structural understanding of this subclass.

Findings

Characterization of interval nest digraphs via nest orderings.

Identification of forbidden patterns in vertex orderings.

Completes the structural picture of interval digraph subclasses.

Abstract

A digraph consisting of a set of vertices $V$ and a set of arcs $E$ is called an interval digraph if there exists a family of closed intervals $\{I_u,J_u\}_{u \in V}$ such that $uv$ is an arc if and only if the intersection of $I_u$ and $J_v$ is non-empty. Interval digraphs naturally generalize interval graphs, by extending the classical interval intersection model to directed graphs. Several subclasses of interval digraphs have been studied in the literature-such as balanced, chronological and catch interval digraphs-each characterized by admitting interval representations that satisfy specific restrictions. Among these, interval nest digraphs are the ones that admit an interval representation in which $J_u$ is contained in $I_u$ for all vertices $u$ of $V$. In this work, we provide a complete characterization of interval nest digraphs in terms of vertex linear orderings with forbidden patterns, which we call nest orderings. This result completes the picture of vertex-ordering characterizations among the main subclasses of interval digraphs.