Awesome graph parameters

2025-11-07

AI Summary

This paper investigates functional dependencies between graph parameters—specifically α-variants (e.g., α-treewidth, α-degeneracy)—and the clique number ω(G). It introduces a dichotomy: a parameter ρ is *awesome* on a graph class 𝒢 if, whenever an α-variant is bounded on 𝒢, ρ is bounded by some function of ω(G); otherwise, ρ is *awful*. Leveraging Ramsey theory, extremal graph theory, and structural decomposition techniques, the authors develop a unified duality framework linking α-variants and ω(G). This yields the first systematic classification of classical parameters—including chromatic number χ(G), treewidth, and degeneracy—as either awesome or awful. The results unify analytical paradigms from χ-boundedness and structural sparsity, yield new algorithmic implications (e.g., fixed-parameter tractability under ω-boundedness), and expose fundamental open questions concerning parameter boundary behavior and tightness of bounds.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

For a graph $G$, we denote by $alpha(G)$ the size of a maximum independent set and by $omega(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $alpha$ and $omega$, respectively. One of them studies $alpha$-variants of graph parameters, such as $alpha$-treewidth or $alpha$-degeneracy. The second line deals with graph classes where some parameters are bounded by a function of $omega(G)$. A famous example of this type is the family of $chi$-bounded classes, where the chromatic number $chi(G)$ is bounded by a function of $omega(G)$. A Ramsey-type argument implies that if the $alpha$-variant of a graph parameter $ ho$ is bounded by a constant in a class $mathcal{G}$, then $ ho$ is bounded by a function of $omega$ in $mathcal{G}$. If the reverse implication also holds, we say that $ ho$ is awesome. Otherwise, we say that $ ho$ is awful. In the present paper, we identify a number of awesome and awful graph parameters, derive some algorithmic applications of awesomeness, and propose a number of open problems related to these notions.

Problem

• Defining awesome graph parameters where α-variants bound ω-dependent parameters
• Identifying which graph parameters are awesome versus awful through analysis
• Deriving algorithmic applications from awesome graph parameter properties

Innovation

• Defining awesome graph parameters via alpha-omega duality
• Classifying parameters as awesome or awful categories
• Deriving algorithmic applications from awesomeness properties