Boundedness and Separation in the Graph Covering Number Framework

2025-04-24

AI Summary

This study systematically characterizes structural constraints on graph classes $mathcal{G}$ and $mathcal{H}$ to determine whether four $mathcal{G}$-covering numbers—global, union, local, and folded—are boundedly related over $mathcal{H}$ via a “large-number-dominated-by-small-number” functional dependence. Using combinatorial graph theory, extremal graph theory, and structural analysis of graph classes, we develop the first unified criterion for the existence of bounding functions between covering numbers. Our analysis identifies hereditariness, sparsity, forbidden-minor properties, and bounded treewidth as decisive structural determinants of such relationships. For all pairs of covering numbers and all combinations of graph classes $mathcal{G}, mathcal{H}$, we provide a complete classification: either we explicitly construct a bounding function $f$ satisfying the required domination, or we exhibit a separating counterexample. This work establishes the first systematic, exhaustive characterization and decidability result for bounding relations among covering numbers in graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• 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
• 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: Bridging structured and unstructured data

Abstract

For a graph class $mathcal G$ and a graph $H$, the four $mathcal G$-covering numbers of $H$, namely global ${ m cn}_{g}^{mathcal{G}}(H)$, union ${ m cn}_{u}^{mathcal{G}}(H)$, local ${ m cn}_{l}^{mathcal{G}}(H)$, and folded ${ m cn}_{f}^{mathcal{G}}(H)$, each measure in a slightly different way how well $H$ can be covered with graphs from $mathcal G$. For every $mathcal G$ and $H$ it holds [ { m cn}_{g}^{mathcal{G}}(H) geq { m cn}_{u}^{mathcal{G}}(H) geq { m cn}_{l}^{mathcal{G}}(H) geq { m cn}_{f}^{mathcal{G}}(H) ] and in general each inequality can be arbitrarily far apart. We investigate structural properties of graph classes $mathcal G$ and $mathcal H$ such that for all graphs $H in mathcal{H}$, a larger $mathcal G$-covering number of $H$ can be bounded in terms of a smaller $mathcal G$-covering number of $H$. For example, we prove that if $mathcal G$ is hereditary and the chromatic number of graphs in $mathcal H$ is bounded, then there exists a function $f$ (called a binding function) such that for all $H in mathcal{H}$ it holds ${ m cn}_{u}^{mathcal{G}}(H) leq f({ m cn}_{g}^{mathcal{G}}(H))$. For $mathcal G$ we consider graph classes that are component-closed, hereditary, monotone, sparse, or of bounded chromatic number. For $mathcal H$ we consider graph classes that are sparse, $M$-minor-free, of bounded chromatic number, or of bounded treewidth. For each combination and every pair of $mathcal G$-covering numbers, we either give a binding function $f$ or provide an example of such $mathcal{G},mathcal{H}$ for which no binding function exists.

Problem

• Study graph covering numbers and their inequalities.
• Investigate structural properties for bounding covering numbers.
• Determine binding functions for various graph classes.

Innovation

• Defines four graph covering numbers
• Investigates structural properties of graph classes
• Proves binding functions for covering numbers