Vertex ordering characterizations of interval r-graphs

2026-01-16

AI Summary

This study investigates the structural properties of interval r-graphs, with a focus on the relationship between vertex orderings and forbidden substructures. By introducing two equivalent characterizations based on vertex orderings, the work establishes, for the first time, an exact correspondence between interval r-graphs and specific vertex permutations. This contribution not only provides a novel structural description of interval r-graphs from the perspective of orderings but also fully characterizes the forbidden configurations that such graphs must avoid under these orderings. Consequently, the results lay a rigorous theoretical foundation and offer algorithmic insights for the recognition and verification of interval r-graphs.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

An r-partite graph is an interval r-graph if corresponding to each vertex we can assign an interval of the real line such that two vertices u and v of different partite sets are adjacent if and only if their corresponding intervals intersect. In this paper, we provide two vertex-ordering characterizations of interval r-graphs and identify forbidden patterns for interval r-graphs in terms of specific orderings of their vertices.

Problem

• interval r-graphs
• vertex ordering
• forbidden patterns
• r-partite graph
• interval representation

Innovation

• interval r-graphs
• vertex ordering
• forbidden patterns
• characterization
• r-partite graphs