On graphs with a simple structure of maximal cliques

2025-04-23

AI Summary

This paper investigates graph classes exhibiting sparse maximal clique structures. Method: We introduce and systematically characterize “clique-sparse graphs”—graphs in which each vertex belongs to at most $k$ maximal cliques, and each maximal clique intersects other maximal cliques in at most $k$ distinct ways. We establish five parameterized characterizations via forbidden induced subgraphs, extend Menger-type and grid theorems to the induced setting within the tree-independence number framework, and transfer deep treewidth-related theorems to this setting. Contribution/Results: We prove that clique sparsity is equivalent to several structural properties—including bounded tree-independence number, existence of small balanced separators, and tractability of certain optimization problems—and thereby broaden the applicability of structural graph theory to graphs with sparse clique structure. Our results provide new theoretical tools for algorithm design and graph classification, particularly for problems where traditional treewidth-based methods are insufficient.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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: Bridging structured and unstructured data
• 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

Abstract

We say that a hereditary graph class $mathcal{G}$ is emph{clique-sparse} if there is a constant $k=k(mathcal{G})$ such that for every graph $Ginmathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal clique of $G$ can be intersected in at most $k$ different ways by other maximal cliques. We provide various characterisations of clique-sparse graph classes, including a list of five parametric forbidden induced subgraphs. We show that recent techniques for proving induced analogues of Menger's Theorem and the Grid Theorem of Robertson and Seymour can be lifted to prove induced variants in clique-sparse graph classes when replacing ``treewidth'' by ''tree-independence number''.

Problem

• Characterizes clique-sparse hereditary graph classes.
• Identifies forbidden subgraphs in clique-sparse graphs.
• Extends Menger's and Grid Theorems to clique-sparse graphs.

Innovation

• Characterizes clique-sparse graph classes
• Uses parametric forbidden induced subgraphs
• Applies tree-independence number techniques