## 2 Bunch Decomposition Technique for Planar Graphs  
In this paper, bunches and related counting methods are the basic techniques used to solve the problem. This section presents the technique of dividing planar graphs into bunches, which has been modified compared to the original work. It should be noted that the division into bunches is used only as a theoretical tool when proving the approximation factor. The approximation algorithm itself does not use this division. However, without this technique, it would be harder to prove the approximation factor in such a simple and basic way. In our approach, we use the concept of _bunches_, introduced in [18]. In this article, we present this technique, but it has been slightly modified for proper use. This technique works well with existing algorithms and shows an effective way to divide planar graphs into subgraphs with diameter of at most three. Additionally, the estimates useful for triangle-free graphs are proven differently than in the original work. In the rest of the article, we assume that we have a graph \(G=(V,E)\) with its planar embedding \(\sigma:G\to\mathbb{R}^{2}\). Let us denote the surface created by this embedding as \(S_{g}\). Note that the planar embedding is required only for analysis, while the algorithm itself does not use this information. To simplify the presentation, we focus on planar graphs, although our approach could be easily extended to graphs with bounded genus with similar approximation factors. Let us move on to the formal definition of bunches. **Definition 1**. : _Let \(P_{1}=v_{1}v_{2},\ldots,v_{k}\) and \(P_{2}=w_{1}w_{2},\ldots,w_{l}\) be two paths in a graph \(G\). We say that the paths \(P_{1}\) and \(P_{2}\)**cross**  each other if there is a common vertex \(v\in V(P_{1})\setminus\{v_{1},v_{k}\}\) such that for arbitrarily small \(\epsilon>0\) we find that \(\sigma(P_{2})\) contains points from more than one component of \(C(\sigma(v),\epsilon)\setminus\sigma(P_{1})\), where \(C(\sigma(v),\epsilon)\) is the disk at \(v\) with radius \(\epsilon\), and this disk is cut along the path \(P_{1}\). In this case, we say that \(P_{1}\) and \(P_{2}\)**cross at vertex** \(v\)._  
Figure 1 contains an example of _crossing paths_. Now, we are ready to define a set of special paths \(\mathcal{P}\). **Definition 2**. : _Let \(X\subset V(G)\) be a set of vertices. We say that \(\mathcal{P}\) is a **set of \(\boldsymbol{X}\) special paths** if the following conditions hold:_  
1. _Each path_ \(P\in\mathcal{P}\) _starts and ends at vertices in_ \(X\)_, with the condition that_ \(P\) _does not start and end at the same vertex._
2. _The paths in_ \(\mathcal{P}\) _do not cross each other, although they may share common vertices or even edges (see Figure_ 1_)._
3. _The internal vertices of each path_ \(P\in\mathcal{P}\) _do not belong to_ \(X\)_._  
_Every \(P\in\mathcal{P}\) is referred to as an \(\boldsymbol{X}\)**special path**._  
The following lemmas are straightforward and analogous to those presented in our previous work. However, to make this paper complete and independent, we provide their proofs here. Now, we are prepared to introduce the concept of a bunch. **Definition 3**. : _Let \(B\subseteq\mathcal{P}\) be a subset of all \(X\) special paths. We define \(B\) as an \(\boldsymbol{X}\)**bunch** (or simply a **bunch**) if the following conditions hold:_  
1. _There exist vertices_ \(v,w\in X\) _(with_ \(v\
eq w\)_), such that_ \(v\) _and_ \(w\) _are the endpoints of every special path in_ \(B\)_._
2. \(B\) _is maximal with respect to inclusion._
3. _There exists_ \(S\subseteq S_{g}\) _homeomorphic to the unit disk containing_ \(B\)_, where_ \(S\) _intersects_ \(X\) _only at vertices_ \(v\) _and_ \(w\)_._  
_Moreover, two special paths \(P_{1},P_{2}\in B\) (not necessarily disjoint or even distinct), such that \(F:=\sigma(P_{1})\cup\sigma(P_{2})\) and the interior region of \(F\) contains all special paths from \(B\), are called the **boundary paths of the bunch**\(B\) and are denoted as \(\text{Bound}(B)\).