The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well-known that minimum dominating sets cannot be well approximated locally on general graphs, over the last years, there has been much progress on computing good local approximations on sparse graphs, and in particular on planar graphs. In this paper we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and (2) a local O(log∗ n)-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments, but on combinatorial density arguments only.

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