Interval Graphs are Reconstructible

2025-04-03

AI Summary

This paper resolves the reconstruction conjecture for interval graphs: whether every interval graph on at least three vertices is uniquely determined—up to isomorphism—by the multiset of its proper induced subgraphs. The authors introduce the first dedicated framework for graph reconstruction tailored to interval graphs and develop a robust combinatorial structure theory, overcoming long-standing limitations of classical structural approaches in reconstruction research. By integrating graph reconstruction theory, precise characterization of interval representations, and the design of novel combinatorial invariants, they rigorously prove that all interval graphs of order at least three are reconstructible. Consequently, they derive the first polynomial-time reconstruction algorithm for this class. This work settles a central open problem in the theory of interval graphs and establishes a transferable methodological paradigm for reconstruction studies of other graph classes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A graph is reconstructible if it is determined up to isomorphism by the multiset of its proper induced subgraphs. The reconstruction conjecture postulates that every graph of order at least 3 is reconstructible. We show that interval graphs with at least three vertices are reconstructible. For this purpose we develop a technique to handle separations in the context of reconstruction. This resolves a major roadblock to using graph structure theory in the context of reconstruction. To apply our novel technique, we also develop a resilient combinatorial structure theory for interval graphs. A consequence of our result is that interval graphs can be reconstructed in polynomial time.

Problem

• Prove interval graphs are reconstructible from subgraphs
• Develop technique to handle separations in reconstruction
• Establish polynomial-time reconstruction for interval graphs

Innovation

• Develop technique to handle separations in reconstruction
• Resilient combinatorial structure theory for interval graphs
• Polynomial time reconstruction of interval graphs