arXiv:2610.38020v2 Announce Type: replace Abstract: It was recently shown (arXiv:2610.09618) that the infimum of the approximation ratios of deterministic constant-round LOCAL algorithms for minimum dominating set on planar graphs is 7. The $(7+\varepsilon)$-approximation given there runs in $T_0+O((c_0/\varepsilon)^{\kappa})$ rounds with $\kappa=\ln 3/\ln(24/23)\approx 25.81$, and it is asked there whether this exponent can be substantially reduced. We replace the heaviest-edge star selection in its clustering step by a random-centre rule: every part becomes a centre independently with probability $p$, and every non-centre joins its heaviest centre neighbour. For planar graphs this gives $\kappa<5.89$ with $p=3/8$ and $\kappa<6.7$ with $p=1/2$. For every proper minor-closed class of density $a$, the new exponent is never worse, and with about $\frac12\log_2 a$ fair coins per part and level it is smaller by a factor that tends to $8$ as the density grows. The planar bounds hold verbatim on the torus, the projective plane and the Klein bottle. We then turn to the dependence on $\varepsilon$. On paths, every deterministic $r$-round algorithm for fractional domination, and every randomized one that always outputs a dominating set, has (expected) ratio at least $1+1/(r+4)$. So these approximations need $\Omega(1/\varepsilon)$ rounds on every minor-closed class that contains all paths. For fractional domination on paths of known length this is tight up to an additive constant. On forests and on outerplanar graphs, with no bound on the degree, $O(1/\varepsilon)$ rounds suffice, so there the optimal number of rounds is $\Theta(1/\varepsilon)$. The planar level bounds, the bounds on paths and the deterministic bound on forests are machine-checked in Lean 4 with Mathlib. Other methods reach ratio $7+\varepsilon$ in fewer rounds against the integral optimum (arXiv:2610.31203) and ratio $7$ in a constant number of rounds (arXiv:2611.02057).