## 1 Introduction
[PERSON] also presented a local (constant-time) algorithm that provides a 694-approximation of MDS on planar graphs, using messages of length at most \(O(\log n)\) and operating in the port numbering model [21]. Constant-factor approximations for planar graphs were later extended to classes of graphs with bounded genus [22; 23], sublogarithmic expansion [24], and sparse graphs [25]. Regarding lower bounds, [PERSON] et al. [26] established that there is no deterministic local algorithm (distributed graph algorithm with constant running time) that achieves a \((7-\epsilon)\)-approximation of the minimum dominating set on planar graphs for any positive constant \(\epsilon\). The most recent significant improvement was presented by [PERSON] et al. [27]. These authors developed a LOCAL algorithm achieving a \((11+\epsilon)\)-approximation for any \(\epsilon>0\) on planar graphs, significantly narrowing the gap between upper and lower bounds. For triangle-free graphs (i.e., graphs that do not contain a cycle of length 3), the MDS problem remains NP-hard. However, the special structure of these graphs allows for the development of efficient approximation algorithms. In contrast to these sequential results, distributed algorithms for the same problem typically achieve worse approximation ratios due to their limited access to global information. In the distributed setting, the previous best approach for triangle-free graphs achieved an approximation ratio of 32 [28]. Recently, [PERSON] et al. [27] improved this result, achieving an approximation ratio of \((8+\epsilon)\) for any \(\epsilon>0\) in a paper published in 2025. In this paper, we present a new distributed approximation algorithm for the MDS problem in planar triangle-free graphs, which achieves an approximation ratio of 6 in constant time. While this is worse than the best known sequential approximation of 5, it represents a significant improvement in the distributed setting. Our algorithm uses the \"bunch\" technique, allowing for efficient estimation of the size of the returned dominating set. A key distinction can be made between our algorithm and the recent work by [PERSON] et al. Their algorithm works in three steps: first, they pick vertices \(v\) such that \(\mu(N[v])>k\) for some small constant \(k\) (e.g., \(k=2\)), where \(N[v]\) denotes the closed neighborhood of \(v\) and \(\mu(N[v])\) represents the minimum size of a set that dominates \(N[v]\). Next, they build special sequences of vertices that cover many others, and, finally, they use linear programming to cover any remaining vertices. It is important to note that their method is a general algorithm that can be applied to any planar graph with an approximation ratio of \((11-\epsilon)\), not just triangle-free graphs. Our algorithm takes a different approach to selecting vertices. We use the special bunch structure of triangle-free planar graphs rather than their more general method. This approach helps us to obtain a better approximation ratio (6 instead of \(8+\epsilon\)) while still running in constant time in the LOCAL model. Our algorithm is also easier to understand and implement because it directly selects vertices based on their degree, instead of using complex sequences and linear programming methods. The structure of this paper is as follows. In Section 2, we introduce the bunch decomposition technique, which partitions planar graphs into smaller components with bounded diameter. We present several key lemmas about bunch properties and establish bounds on the number of external vertices, which are crucial for our approximation ratio analysis. In Section 3, we describe our distributed approximation algorithm that constructs a dominating set in two efficient phases: first handling vertices of degree 2, then optimizing the selection of the remaining vertices.