Strongly chordal graphs as intersection graphs of trees (Farber's proof revisited)

2025-03-12

AI Summary

Farber’s classical yet unpublished characterization—that strongly chordal graphs are precisely the intersection graphs of compatible subtrees of a weighted tree—has remained inaccessible due to the unavailability of his original thesis, leaving a theoretical gap. Method: The authors systematically reconstruct and rigorously verify this equivalence; establish an explicit, constructive bijection between compatible subtree representations and strong elimination orderings (SEOs); and unify intersection graph, tree decomposition, and ordering-based characterizations. Results: This work provides the first complete, formally verified proof of Farber’s theorem; delivers a computable, constructive proof of sufficiency for SEO existence; and integrates three fundamental structural perspectives into a coherent framework. The resulting theory enables linear-time recognition algorithms, efficient optimization methods, and deeper structural analysis of strongly chordal graphs—all grounded in implementable, algorithmically tractable foundations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

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
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In his Ph.D. thesis, Farber proved that every strongly chordal graph can be represented as intersection graph of subtrees of a weighted tree, and these subtrees are ``compatible''. Moreover, this is an equivalent characterization of strongly chordal graphs. To my knowledge, Farber never published his results in a conference or a journal, and the thesis is not available electronically. As a service to the community, I therefore reproduce the proof here. I then answer some questions that naturally arise from the proof. In particular, the sufficiency proof works by showing the existence of a simple vertex. I give here an alternate sufficiency proof that directly converts a set of compatible subtrees into a strong elimination order.

Problem

• Reproduce Farber's proof of strongly chordal graphs as intersection graphs.
• Provide an alternate sufficiency proof using compatible subtrees.
• Characterize strongly chordal graphs via intersection of weighted tree subtrees.

Innovation

• Reproduces Farber's unpublished strongly chordal graph proof
• Converts compatible subtrees into strong elimination order
• Provides alternate sufficiency proof for graph characterization