Characterization of sparse monotone graph classes with bounded domination-to-2-independence ratio
Marthe Bonamy, Zdeněk Dvořák, Lukas Michel, David Mikšaník
TL;DR
The paper addresses bounding the domination number $\gamma(G)$ by the $2$-independence number $\alpha_2(G)$ in monotone graph classes with bounded average degree. It introduces cascades and their slope as the central structural obstruction, and proves an exact characterization: a class has a bounded $\gamma/\alpha_2$ ratio iff it contains no cascades of unbounded slope. The authors develop a toolkit of $m$-cascades, foundations, and probabilistic rounding lemmas, together with Ramsey-type arguments, to derive both qualitative and quantitative bounds. Applications include classes with maximum average degree below 6 and certain minor-closed families, as well as a Ramsey-based bound for $K_k$-subdivision-free classes, establishing linear domination bounds in terms of $\alpha_2$ for these sparse graph classes.
Abstract
We give an exact characterization of monotone graph classes C with bounded average degree that satisfy the following property: The domination number of every graph from C is bounded by a linear function of its 2-independence number.