Recognizing Leaf Powers and Pairwise Compatibility Graphs is NP-Complete

2025-10-22

AI Summary

This paper resolves the long-standing open problem of the computational complexity of recognizing leaf power graphs and pairwise compatibility graphs (PCGs), establishing that both recognition problems are NP-complete. The proof employs a constructive polynomial-time reduction, which systematically extends the hardness result to broader graph classes—including multi-interval PCGs—within a hierarchical framework. By integrating structural graph-theoretic analysis with tools from computational complexity theory, the work demonstrates the inherent intractability of recognizing graph classes naturally arising from phylogenetic modeling. This result definitively settles a decade-old conjecture regarding the complexity of PCG recognition and provides a fundamental characterization of the theoretical limits of biologically inspired graph structures. It thereby advances foundational understanding at the intersection of computational graph theory and systems biology.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models

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: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Leaf powers and pairwise compatibility graphs were introduced over twenty years ago as simplified graph models for phylogenetic trees. Despite significant research, several properties of these graph classes remain poorly understood. In this paper, we establish that the recognition problem for both classes is NP-complete. We extend this hardness result to a broader hierarchy of graph classes, including pairwise compatibility graphs and their generalizations, multi interval pairwise compatibility graphs.

Problem

• Proving NP-completeness of leaf power recognition
• Establishing NP-completeness for pairwise compatibility graphs
• Extending hardness results to generalized graph hierarchies

Innovation

• Proves leaf power recognition is NP-complete
• Shows pairwise compatibility graphs are NP-complete
• Extends hardness to multi interval compatibility graphs