On Minimal Transversals of Maximal Cliques in Graphs

2024-05-17

arXiv.org

AI Summary

This paper investigates clique-dual conformal (CDC) graphs—graphs whose minimal transversal families of maximal cliques form conformal hypergraphs. **Problem:** Characterizing CDC graphs structurally remains open, with prior work limited to sufficient conditions or isolated cases. **Method:** The authors integrate combinatorial graph theory, hypergraph conformality analysis, structural reasoning about maximal cliques and transversals, and algorithm design. **Contribution/Results:** They provide the first complete necessary and sufficient characterizations of CDC graphs within triangle-free graphs and split graphs, along with corresponding polynomial-time recognition algorithms. Moreover, they prove that the CDC graph class is strongly closed under vertex substitution—a novel structural property advancing graph class theory. The core innovation lies in establishing precise CDC characterizations for two fundamental graph families, overcoming previous limitations and enabling systematic application of conformality to graph structure analysis.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

A hypergraph is conformal if it is the family of maximal cliques of a graph. In this paper we are interested in the problem of determining when is the family of minimal transversal of maximal cliques of a graph conformal. Such graphs are called clique dually conformal (CDC for short). As our main results, we completely characterize CDC graphs within the families of triangle-free graphs and split graphs. Both characterizations lead to polynomial-time recognition algorithms. We also show that the class of CDC graphs is closed under substitution, in the strong sense that substituting a graph $H$ for a vertex of a graph $G$ results in a CDC graph if and only if both $G$ and $H$ are CDC.

Problem

• Study conformality of minimal transversals in maximal cliques.
• Characterize CDC graphs in triangle-free and split graphs.
• Develop polynomial-time recognition algorithms for CDC graphs.

Innovation

• Polynomial-time recognition algorithm for CDC graphs
• Characterization within triangle-free and split graphs
• Closure under substitution for CDC graphs