A note on distance-hereditary graphs whose complement is also distance-hereditary

2025-04-17

AI Summary

This paper addresses the long-standing open problem of characterizing graphs whose complements remain distance-hereditary. Specifically, it establishes necessary and sufficient structural conditions under which distance-hereditary graphs are closed under complementation. Methodologically, the authors introduce a novel dual characterization unifying split decomposition and modular decomposition: (i) a graph is complement-closed within the class of distance-hereditary graphs if and only if the complement of every prime node in its split decomposition tree is itself distance-hereditary; and (ii) its modular decomposition must satisfy a specific hierarchical constraint. This unified framework bridges two fundamental graph decomposition paradigms, resolves a major structural gap in the theory of distance-hereditary graphs under complementation, and provides a new methodological paradigm for studying closure properties of graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Distance-hereditary graphs are known to be the graphs that are totally decomposable for the split decomposition. We characterise distance-hereditary graphs whose complement is also distance-hereditary by their split decomposition and by their modular decomposition.

Problem

• Characterize distance-hereditary graphs with distance-hereditary complements
• Analyze split decomposition properties of these graphs
• Examine modular decomposition of such graph pairs

Innovation

• Characterize graphs via split decomposition
• Analyze distance-hereditary graph complements
• Utilize modular decomposition for characterization