## 1 Introduction  
The minimum dominating set (MDS) problem is a fundamental optimization problem in graph theory, playing a crucial role in both theoretical computer science and practical applications [1, 2]. For a graph \(G=(V,E)\), a dominating set is a subset \(D\subseteq V\) such that every vertex \(v\in V\) either belongs to \(D\) or is adjacent to at least one vertex in \(D\). A minimum dominating set is a dominating set of minimum cardinality. It is important to note that there is a significant difference between sequential algorithms, which process the entire graph with complete information, and distributed algorithms, which operate with limited local information at each node. In the context of solving the MDS problem, distributed algorithms that can be efficiently implemented in networks are particularly interesting. The LOCAL model, first introduced by [PERSON] [3], is one of the basic models of distributed computing, where a network is represented as a graph with vertices corresponding to processors and edges representing communication channels. In this model, in each round every vertex can send any message to all its neighbors, perform local computations, and receive messages from its neighbors. The complexity of an algorithm in the LOCAL model is measured by the number of communication rounds needed to solve the problem. An algorithm is called local if it operates in a constant number of rounds, regardless of the network size. The computational complexity of the MDS problem has been studied in various computational models and for numerous graph classes. In general, the problem is NP-hard [4], and, furthermore, finding an approximation with a factor of \((C\log\Delta(G))\) for some constant \(C\) is also NP-hard [5]. A simple but effective greedy strategy achieves a logarithmic approximation ratio of \(\ln n\), where \(n\) is the number of vertices in the graph [6, 7]. Importantly, this approximation ratio is essentially the best possible for arbitrary graphs, as obtaining an approximation ratio better than \((1-\epsilon)\ln n\) (for any \(\epsilon>0\)) is NP-hard [8]. Researchers have sought restricted graph classes where better approximations are possible. For graph classes with subexponential expansion [9], the problem admits a polynomial-time approximation scheme (PTAS). Constant-factor approximations exist for graphs with bounded degeneracy [10; 11], bounded VC-dimension [12; 13], and biclique-free graphs [14] achieve an \(\mathcal{O}(\ln\gamma)\) approximation ratio. For graphs with bounded arboricity, which includes planar and triangle-free planar graphs, significant results in sequential algorithms have been achieved. [PERSON] and [PERSON] [15] provided a sequential algorithm achieving a \(3d\)-approximation, where \(d\) is the arboricity of the graph. Later, [PERSON] [16] improved this result to a \((2d+1)\)-approximation factor. Since planar triangle-free graphs have arboricity of at most 2, these results yield a sequential algorithm with an approximation factor of at most 5. While these sequential algorithms achieve excellent approximation ratios, our focus in this paper is on distributed algorithms, which face the additional constraints of limited local information and communication requirements. In the distributed setting, [PERSON] et al. [17] proved lower bounds on the approximation of dominating sets depending on the number of communication rounds. Their results imply that constant-factor approximations essentially require \(\Omega(\sqrt{\log n/\log\log n})\) rounds. For specific graph classes, however, better results are possible. For planar graphs, the MDS problem becomes more tractable. [PERSON] et al. [18] achieved constant-factor approximations in \(O(\log^{*}n)\) rounds on planar graphs. [PERSON] et al. proposed a constant-round algorithm that computes a 130-approximation of the minimum dominating set in planar graphs [19]. A subsequent careful analysis by [PERSON] [20] showed that the same algorithm actually provides a 52-approximation.