On Threshold Compatibility Graphs

2026-04-21

AI Summary

This work addresses the limited expressive power of classical pairwise compatibility graphs (PCGs) by introducing a unified $(k,t)$-threshold PCG framework, wherein adjacency between two vertices is defined by the acceptance of at least $t$ out of $k$ underlying PCG predicates. This formulation generalizes existing extensions such as $k$-interval PCGs and $k$-OR/AND-PCGs. The study establishes fundamental expressiveness bounds: every $n$-vertex graph is a $(n,t)$-threshold PCG for appropriate $t$, yet for any fixed $(k,t)$, such graphs are asymptotically rare among all graphs. Moreover, the hierarchy of $k$-AND-PCGs is shown to be strictly increasing and not closed under complementation, and a strict separation in expressive power is demonstrated between $k$-AND-PCGs and $k$-interval PCGs. These results are derived through graph-theoretic constructions, asymptotic analysis, and carefully crafted counterexamples, yielding a systematic characterization of structural and closure properties within this class.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

Pairwise Compatibility Graphs (PCGs) form a tree-metric graph class that originated in phylogeny and has since attracted sustained interest in graph theory. Several natural generalizations have been proposed in order to overcome the expressive limitations of classical PCGs, including $k$-interval-PCGs, $k$-OR-PCGs, and $k$-AND-PCGs. In this paper, we introduce $(k,t)$-threshold-PCGs, a threshold-based framework that unifies these generalized notions: adjacency is determined by whether at least $t$ among $k$ underlying PCG predicates accept the vertex pair. We investigate the expressive power of this model from both constructive and asymptotic viewpoints. On the positive side, we show that every graph on $n$ vertices is a $(n,t)$-threshold-PCG for every $1 \le t \le n$. On the negative side, we prove that for every fixed pair $(k,t)$, the class of $(k,t)$-threshold-PCGs is asymptotically rare among all graphs. As a consequence, we obtain sharp separations from previously studied models, including a strict expressive gap relative to $k$-interval-PCGs. We also study explicit obstruction families through incidence graphs and derive additional structural consequences for the conjunction case, including the strictness of the $k$-AND-PCG hierarchy and the failure of closure under complement.

Problem

• Pairwise Compatibility Graphs
• expressive power
• threshold-based framework
• graph classes
• generalized PCGs

Innovation

• threshold-PCG
• graph expressiveness
• pairwise compatibility graphs
• k-AND-PCG hierarchy
• asymptotic rarity