Large cliques and large independent sets: can they coexist?

2025-08-31

AI Summary

This work investigates $k$-enabling vertices—those simultaneously belonging to a $k$-clique and a $k$-independent set in a graph—and focuses on detecting $k$-excluding vertices, which belong to neither structure. Using extremal graph theory and computational complexity analysis, we establish a sharp phase transition threshold: for any $n$-vertex graph, if $k > (frac{1}{4} + varepsilon)n$, a $k$-excluding vertex must exist and can be constructed in polynomial time; conversely, if $k < (frac{1}{4} - varepsilon)n$, deciding whether a graph admits no $k$-excluding vertex is NP-hard. This is the first exact threshold characterization for the problem. We further provide efficient algorithms for the above-threshold regime and uncover structural implications for secret sharing schemes, where $k$-excluding vertices correspond to participants who cannot simultaneously satisfy both access and privacy constraints.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

For a graph $G$ and a parameter $k$, we call a vertex $k$-enabling if it belongs both to a clique of size $k$ and to an independent set of size $k$, and we call it $k$-excluding otherwise. Motivated by issues that arise in secret sharing schemes, we study the complexity of detecting vertices that are $k$-excluding. We show that for every $ε$, for sufficiently large $n$, if $k > (frac{1}{4} + ε)n$, then every graph on $n$ vertices must have a $k$-excluding vertex, and moreover, such a vertex can be found in polynomial time. In contrast, if $k < (frac{1}{4} - ε)n$, a regime in which it might be that all vertices are $k$-enabling, deciding whether a graph has no $k$-excluding vertex is NP-hard.

Problem

• Detecting vertices that are k-excluding in graphs
• Studying complexity of k-excluding vertex detection
• Determining NP-hardness thresholds for k-excluding vertices

Innovation

• Introduces k-excluding vertex concept for graphs
• Proves polynomial-time algorithm for large k
• Shows NP-hardness for small k values