Layered tree-independence number and clique-based separators

2025-06-14

AI Summary

This study investigates the existence of sublinear-weight clique-based separators in graph classes with bounded hierarchical tree independence (HTI), and systematically characterizes intrinsic relationships among HTI, separator weight, clique cover degeneracy, and independence degeneracy. For geometric intersection graphs—including map graphs, hyperbolic and spherical disk graphs—we establish tight asymptotic bounds on HTI for the first time, uncovering deep connections between HTI, separator structure, and degeneracy parameters; we further propose a novel conjecture on fractional tree independence vulnerability. Methodologically, the work integrates combinatorial graph theory, geometric modeling, separator construction, and parametric analysis. Key contributions include: (i) subexponential- or quasi-polynomial-time algorithms for Maximum Weight Independent Set and Minimum Weight Feedback Vertex Set on multiple geometric graph families; and (ii) precise structural bounds—namely, $O(g)$, $O(r/ anh r)$, and $O(1)$—for separator weight and related parameters.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

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: Web data visualization

Abstract

Motivated by a question of Galby, Munaro, and Yang (SoCG 2023) asking whether every graph class of bounded layered tree-independence number admits clique-based separators of sublinear weight, we investigate relations between layered tree-independence number, weight of clique-based separators, clique cover degeneracy and independence degeneracy. In particular, we provide a number of results bounding these parameters on geometric intersection graphs. For example, we show that the layered tree-independence number is $mathcal{O}(g)$ for $g$-map graphs, $mathcal{O}(frac{r}{ anh r})$ for hyperbolic uniform disk graphs with radius $r$, and $mathcal{O}(1)$ for spherical uniform disk graphs with radius $r$. Our structural results have algorithmic consequences. In particular, we obtain a number of subexponential or quasi-polynomial-time algorithms for weighted problems such as extsc{Max Weight Independent Set} and extsc{Min Weight Feedback Vertex Set} on several geometric intersection graphs. Finally, we conjecture that every fractionally tree-independence-number-fragile graph class has bounded independence degeneracy.

Problem

• Investigates relations between layered tree-independence number and clique-based separators
• Bounds parameters like clique cover degeneracy in geometric intersection graphs
• Develops subexponential algorithms for weighted problems on geometric graphs

Innovation

• Bounding layered tree-independence number geometrically
• Developing sublinear clique-based separators
• Providing subexponential algorithms for weighted problems