arXiv:2610.31203v1 Announce Type: new Abstract: We study distributed approximation of Minimum Dominating Set in a very general class of sparse graphs, extending near-optimal approximation results from the minor-free setting to bounded expansion. In more detail, for fixed $d$ and $\varepsilon>0$, we give a new deterministic constant-round LOCAL algorithm for Minimum Dominating Set on graphs of maximum average degree at most $d$ and bounded depth-one shallow-minor density. Its approximation ratio is $d+1+\varepsilon$, which is optimal up to $\varepsilon$ for every even integer $d\geq 2$. This improves on the recent results of Amiri and Siebertz by extending beyond proper minor-closed classes. From this algorithm, we derive almost-tight constant-round approximations for a number of graph classes. We get $(7+\varepsilon)$-approximations for planar graphs and graphs of bounded Euler genus, and $(5+\varepsilon)$-approximations for their triangle-free subclasses. This answers the $7+\varepsilon$ question of Wawrzyniak positively, and matches the recent results of Amiri and Siebertz. We also apply our algorithm to classes of bounded queue-number or stack-number, obtaining bounds that are tight up to $\varepsilon$. Additionally, we answer questions of Amiri and Siebertz about improving the dependence of the round complexity on $1/\varepsilon$ and obtaining polynomial local computation for fixed parameters, when comparing the output to the usual domination number.