## 2 Bunch Decomposition Technique for Planar Graphs
The minimal surface \(\mathcal{S}_{B}\), such that \(B\subset\mathcal{S}_{B}\) and \(\mathcal{S}_{B}^{in}:=\mathcal{S}_{B}\setminus(P_{1}\cup P_{2})\), we call the **surface of the bunch**\(B\) and the **interior of surface of the bunch**\(B\), respectively._  
**Lemma 1**. : _Let \(\mathcal{P}\) be a set of special paths and \(\mathcal{B}\) be the set of all bunches of paths from \(\mathcal{P}\). Then, every special path \(P\in\mathcal{P}\) belongs to at least one bunch \(B\in\mathcal{B}\)._  
Figure 1: Examples of crossing paths: (a) paths cross each other (and, therefore, do not belong to one set of special paths), (b) paths \(v_{1}v_{3}v_{4}v_{5}\), \(v_{2}v_{3}v_{4}v_{5}\), \(v_{1}v_{3}v_{6}v_{7}\) do not cross each other, and so can belong to one set of special paths. Proof. : The embedding \(\sigma\) maps each path \(P\in\mathcal{P}\) to an arc on the surface \(S_{g}\). For each such arc, there is a surface \(S\subset S_{g}\) that contains the arc \(\sigma(P)\) (i.e., \(\sigma(P)\subset S\)). Thus, every special path in \(\mathcal{P}\) is part of some bunch \(B\in\mathcal{B}\). Lemma 2.: _Let \(B\in\mathcal{B}\) be a bunch and let \(P_{1},P_{2}\in B\) be the boundary paths of \(B\). Then, every vertex \(v\in V(B)\setminus(V(P_{1})\cup V(P_{2}))\) is only adjacent to vertices in \(\mathcal{S}_{B}\)._  
Proof. : The embedding \(\sigma\) maps the paths \(P_{1}\cup P_{2}\) to a closed curve on \(S_{g}\), effectively separating the interior region of \(\sigma(P_{1}\cup P_{2})\) from the rest of the surface \(S_{g}\). Since \(P_{1}\) and \(P_{2}\) form the boundary paths, every vertex in \(V(B)\setminus(V(P_{1})\cup V(P_{2}))\) lies within the interior of the region bounded by \(\sigma(P_{1}\cup P_{2})\) and can, therefore, only be adjacent to vertices contained in \(\mathcal{S}_{B}\). Lemma 3.: _Let \(X,Y\subseteq V(G)\) and \(\mathcal{B}\) be the set of all \(X-Y\) bunches. Then, if \(X=Y\) then \(|\mathcal{B}|\leq 3|X|-6\), for \(|X|\geq 3\)._  
Proof. : The proof can be found in [29]. We select a fixed minimum dominating set \(M\) in \(G\). For each vertex \(v\in V(G)\setminus M\) we set \(m(v)\in M\), such that \(m(v)\) is adjacent to \(v\). Note that the minimum dominating set \(M\), the planar embedding \(\sigma\), the surface \(S_{g}\), and the function \(m(v)\) are used only in the theoretical analysis to prove the algorithm's correctness and approximation ratio. These constructions are not required during the actual execution of the algorithm. We now define the set of M-M paths and show that this set is special, in the sense of Definition 2. Definition 4.: Let \(\mathcal{P}_{M}\) be the collection of all paths \(P\) in \(G\) that meet the following criteria:  
1. \(P:=(m(v),v,w,m(w))\), where \(v,w\in V(G)\setminus M\), \(vw\in E(G)\), and \(m(v)\
eq m(w)\),
2. \(P:=(m(v),v,m_{1})\), where \(v\in V(G)\setminus M\), \(m_{1}\in M\), \(vm_{1}\in E(G)\), and \(m_{1}\
eq m(v)\),
3. \(P:=(m_{1},m_{2})\), where \(m_{1},m_{2}\in M\) and \(m_{1}m_{2}\in E(G)\). Figure 2 shows an example graph \(G\) and the corresponding multigraph \(H\) with marked bunches and edges. It is worth noting that special paths between vertices \(m_{2}\) and \(m_{4}\) were divided into two separate bunches due to vertex \(m_{6}\) lying inside the area between these paths. Moreover, it should be observed that there may exist vertices like \(v_{11}\) that do not belong to any bunch. Figure 2: An example of a multigraph \(H\) created from \(G\). Vertices \(m_{i}\) belong to the dominating set \(M\), while the remaining vertices belong to \(V\setminus M\). **Lemma 4**: _The set \(\mathcal{P}_{M}\) consists of M-M special paths._  
Proof. : It is clear that each path \(P\in\mathcal{P}_{M}\) has both endpoints in \(M\) and does not begin and end at the same vertex. Additionally, every internal vertex of \(P\) is in \(V(G)\setminus M\). Therefore, conditions (a) and (c) of Definition 2 are satisfied. Now, assume for contradiction that condition (b) is not met.