A characterization of interval nest digraphs

2026-03-09

AI Summary

This work addresses the important subclass of interval nest digraphs by introducing the notion of a “nested ordering.” It provides the first complete characterization of this graph class through a linear vertex ordering that forbids specific patterns, thereby establishing an equivalence between interval nest digraphs and such orderings. By integrating tools from graph theory, combinatorial structure analysis, and interval representation models, the study achieves a full structural characterization of interval nest digraphs. This result fills a notable gap in the theory of forbidden patterns for vertex orderings among major subclasses of interval digraphs, offering a foundational insight into their combinatorial properties.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Rule Mining & Pattern Mining

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• interval nest digraphs
• vertex ordering
• forbidden patterns
• characterization
• interval digraphs

Innovation

• interval nest digraphs
• vertex ordering
• forbidden patterns
• nest orderings
• interval digraph characterization