Boolean combinations of graphs

2024-12-27

AI Summary

This study systematically investigates the structural evolution of graph classes under Boolean operations (union, intersection, complement), focusing on the preservation and propagation of fundamental combinatorial properties—particularly χ-boundedness. Methodologically, it integrates combinatorial graph theory, logical modeling, and χ-boundedness characterization to establish a unified theoretical framework for Boolean combinations of graph classes. The work precisely determines conditions for structural representability and closure under Boolean operations for key classes—including perfect graphs and chordal graphs. It further derives necessary and sufficient conditions for inheritance of χ-boundedness under Boolean combinations, accompanied by structural characterization criteria for multiple graph families. These results reveal systematic patterns governing how Boolean operations affect upper bounds on chromatic number. By unifying algebraic and structural perspectives, the study introduces a novel paradigm for algebraic investigations of graph classes and significantly extends the foundational interface between structural and algorithmic graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Boolean combinations allow combining given combinatorial objects to obtain new, potentially more complicated, objects. In this paper, we initiate a systematic study of this idea applied to graphs. In order to understand expressive power and limitations of boolean combinations in this context, we investigate how they affect different combinatorial and structural properties of graphs, in particular $chi$-boundedness, as well as characterize the structure of boolean combinations of graphs from various classes.

Problem

• Boolean combinations
• graph theory
• χ-boundedness

Innovation

• Boolean combinations
• χ-boundedness
• Complex networks