Almost Optimal Constant-Round Approximation of Dominating Set in Graph Classes with Excluded Minors

For every fixed proper minor-closed class $\mathscr C$ and every $ε>0$, we give a deterministic LOCAL algorithm that returns a dominating set of size at most $(2a(\mathscr C)+1+ε)γ_f(G)$ on every $G\in\mathscr C$. Here $a(\mathscr C)$ is the supremum edge-to-vertex ratio in~$\mathscr C$, and $γ_f(G)