## 2 Bunch Decomposition Technique for Planar Graphs
In this case, there must exist two paths \(P_{1},P_{2}\in\mathcal{P}_{M}\) that intersect at some internal vertex \(v\in V(P_{1})\cap V(P_{2})\) (see case \"a\" in Figure 1). Since \(v\in V(G)\setminus M\), both paths \(P_{1}\) and \(P_{2}\) must end at the same vertex \(m(v)\), which contradicts the assumption that they cross. For the remainder of this paper, we assume that the optimal dominating set \(M\) has been fixed and that the set \(\mathcal{P}_{M}\) has been constructed. We will use \(\mathcal{B}\) to denote the set of all \(M\)-\(M\) bunches in \(G\). **Definition 5**: _For a graph \(G=(V,E)\) and any subset of vertices \(X\subseteq V\), we define the following operators:_  
\[N(X) :=\{v\in V\setminus X:\exists_{x\in X}\{v,x\}\in E\}\] \[N[X] :=N(X)\cup\ X\]  
_For simplicity of notation, instead of using the notation \(\{v\}\) for a single vertex \(v\), we will simply write \(N(v)\), \(N[v]\), etc._  
**Definition 6**: _For a graph \(G=(V,E)\) and a set of bunches \(\mathcal{B}\), we define the subset of vertices \(T\):_  
\[T:=\{v\in V\setminus M: \exists B_{1},B_{2}\in\mathcal{B},B_{1}\
eq B_{2}\] \[\text{ such that }v\in V(B_{1})\cap V(B_{2})\}.\]  
_The set \(T\) will be called the set of external vertices of bunches \(\mathcal{B}\) or, in short, the external vertices of bunches. Every vertex in \(V\setminus M\) that is not external is called an internal vertex of \(B\)._  
**Definition 7**: _For a given bunch \(B\in\mathcal{B}\) we define \(B^{\mathrm{OUT}}\) as the set of bunch vertices belonging to set \(T\), i.e., \(B^{\mathrm{OUT}}:=V(B)\cap T\). The set of remaining vertices excluding bunch endpoints is denoted as \(B^{\mathrm{IN}}\), i.e., \(B^{\mathrm{IN}}:=V(B)\setminus(B^{\mathrm{OUT}}\cup M)\)._  
**Lemma 5**: \(|T|\leq 6|M|\)_._  
Proof. : From Lemma 3, we know that the total number of bunches is at most \(3|M|-6\) for \(|M|\geq 3\). Furthermore, each bunch has exactly two boundary paths, so in total we have at most \(2(3|M|-6)\) boundary paths. From the definition of special paths in bunches \(\mathcal{B}\), each boundary path can contain at most two vertices belonging to \(V\setminus M\). Since each vertex \(v\in T\) by definition belongs to at least two different bunches, it is counted at least twice in our estimation. Therefore, we obtain the following:  
\[|T|\leq\frac{4(3|M|-6)}{2}\leq 6|M|\]  
which completes the proof. In the case of triangle-free graphs, we can obtain a better estimate for the number of vertices in the set \(T\). This is due to the fact that triangle-free graphs have fewer edges compared to general planar graphs, which directly affects the structure of bunches. **Lemma 6**: _For each face \(f\) of degree 3 in the bunch graph \(H\) of a triangle-free graph \(G\), there are at most two vertices belonging to the set \(T\) that are incident to this face \(f\)._  
Let us consider a face \(f\) of degree 3 in the bunch graph \(H\). This face is bounded by three vertices belonging to the set \(M\), which we denote as \(m_{1}\), \(m_{2}\), and \(m_{3}\). Each edge in graph \(H\) corresponds to a bunch in graph \(G\), so we have three bunches associated with face \(f\). An example of such a degree 3 face is shown in Figure 3, where vertices \(m_{1}\), \(m_{2}\), and \(m_{3}\) form a triangle, and between each pair of vertices from set \(M\) there are two vertices not belonging to set \(M\):  
Note that on each such path there can be at most two vertices from \(V\setminus M\) by the definition of special paths in bunches \(\mathcal{B}\). Moreover, without loss of generality, let us assume that vertex \(v_{1}\) belongs to two bunches on this face. Vertex \(v_{1}\) belongs to the bunch with path \((m_{1},v_{1},v_{2},m_{2})\), so in order to belong to two bunches \(v_{1}\) must be connected to vertex \(v_{5}\) (the dashed line in Figure 3 between \(v_{1}\) and \(v_{5}\)). Note that for such a situation to occur, vertex \(v_{6}\) must be outside face \(f\).