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 $T_0+O((c_0/\varepsilon)^κ)$ rounds with $κ=\ln 3/\ln(24/23)\approx 25.81$, 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 $κ<5.89$ with $p=3/8$ and $κ<6.7$ 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 $Ω(1/\varepsilon)$ 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 $Θ(1/\varepsilon)$. 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).
In the CONGEST model, exact single-source shortest paths (SSSP) with nonnegative weights and exact maximum $st$-flow in directed planar networks have both been solved in $\tilde O(D^2)$ rounds, where $D$ is the hop diameter, while the lower bound is $Ω(D)$. Given the approximate distance labels of a recent preprint (arXiv:2606.31192), which we use as a black box, we solve exact directed SSSP in $\tilde O(D)$ rounds. It rests on a lemma that answers queries from a virtual source with the labels, and on a corrected discretization of the reduction of exact to approximate distances (arXiv:2210.16351). All further bounds of the form $\tilde O(D)$ and $D\cdot n^{o(1)}$ use these labels. Through the reduction of arXiv:2303.00811, SSSP with negative weights, or the detection of a negative cycle, takes $D\cdot n^{o(1)}$ rounds with high probability. We weight the face-disjoint graph of arXiv:2411.11718 so that its distances are dual distances, which turns distance problems in the dual into problems in a planar network of hop diameter $O(D)$. This gives exact maximum $st$-flow in directed planar networks in $\min\{\tilde O(D^2),D\cdot n^{o(1)}\}$ rounds with high probability, and in $\tilde O(D)$ rounds with $s,t$ on a common face. In undirected planar networks, the value of a minimum $st$-cut and a minimum cut take $\tilde O(D)$ rounds. For this we repair the auxiliary graphs of arXiv:2608.09500 at $s$ and $t$, and we show that its recursion, as we read it, can miss the minimum cut, also with exact shortest paths: on a network with $11$ vertices it reports $37$ for a minimum cut of $15$. Two more shortest-path computations per level make it exact. The virtual-source lemma, the weighting of the face-disjoint graph and the corrected discretization are machine-checked in Lean 4 with Mathlib; all other arguments are proved on paper only. Numerical checks are not used in any proof.