Boundedness and Separation in the Graph Covering Number Framework
Miriam Goetze, Peter Stumpf, Torsten Ueckerdt
TL;DR
The paper develops a unified framework of graph covering numbers (global, union, local, folded) $cn_x^{\mathcal{G}}(H)$ to study decompositions of host graphs $H$ into guest graphs from $\mathcal{G}$. It defines binding functions to relate larger and smaller covering numbers under structural restrictions on $\mathcal{G}$ (component-closed, hereditary, monotone, sparse, bounded chi, bounded mad) and $\mathcal{H}$ (bounded chi, mad, $M$-minor-free, bounded treewidth), proving numerous positive boundedness results and constructing several separating counterexamples. Key techniques include reductions to star forests and bipartite subgraphs, tree decompositions, and subgraph restriction lemmas for hereditary/monotone classes, yielding explicit bounds such as $cn_u \le (\mathrm{tw}(H)+1) cn_l$ for component-closed $\mathcal{G}$ with bounded treewidth hosts, and $cn_u \le 2sd$ under sparse hereditary guests with mad bound $d$, $s=cn_l$. The results provide a comprehensive landscape of when the folded, local, union, and global covering numbers are functionally related, with algorithmic implications for parameters like star arboricity and potential extensions to nowhere-dense or WL-dimension settings.
Abstract
For a graph class $\mathcal G$ and a graph $H$, the four $\mathcal G$-covering numbers of $H$, namely global ${\rm cn}_{g}^{\mathcal{G}}(H)$, union ${\rm cn}_{u}^{\mathcal{G}}(H)$, local ${\rm cn}_{l}^{\mathcal{G}}(H)$, and folded ${\rm cn}_{f}^{\mathcal{G}}(H)$, 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 \[ {\rm cn}_{g}^{\mathcal{G}}(H) \geq {\rm cn}_{u}^{\mathcal{G}}(H) \geq {\rm cn}_{l}^{\mathcal{G}}(H) \geq {\rm cn}_{f}^{\mathcal{G}}(H) \] 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 ${\rm cn}_{u}^{\mathcal{G}}(H) \leq f({\rm cn}_{g}^{\mathcal{G}}(H))$. 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.