Characterization of graph classes with bounded domination-packing ratio

Characterize the graph classes \(\mathcal G\) for which there exists a constant \(c_{\mathcal G}\) satisfying \(\gamma(G)/\rho(G)\le c_{\mathcal G}\) for every graph \(G\in\mathcal G\).

Background

The paper studies when the domination number γ(G)\gamma(G) is bounded by a constant multiple of the packing number ρ(G)\rho(G) throughout a graph class. It establishes bounded-ratio results for several classes, including 2-degenerate, asteroidal triple-free, convex, unit-disk, planar, bounded-treewidth, and bounded-twin-width graphs, while giving unbounded-ratio constructions for other classes. A general characterization of all graph classes admitting such a constant remains unresolved.

References

In this work, we have made progress in resolving the following major open problem. Characterize those graph classes $\mathcal G$ for which there is a constant $c_{\mathcal G}$ such that \frac{\gamma(G)}{\rho(G)}\le c_{\mathcal{G} \quad \text{ for each } \quad G\in \mathcal{G} ~~ .

— On graph classes with constant domination-packing ratio  (2503.05562 - Bonamy et al., 7 Mar 2025) in Section 7, Conclusion