## 3 Preliminaries  
We review some of the key ideas from this study in this part. In the beginning, we are going to discuss intuitionistic fuzzy sets (IFSs) and the construction of an IFS from a fuzzy set. Given an ordinary finite non-empty set \(X\) and \(A\subseteq X\), an Atanassov IFS on \(X\) is a set of the form \(A_{I}=\ \ \{(x,\mu_{A}(x),\
u_{A}(x))|x\in X\}\) wherein \(\mu_{A},\
u_{A}:X\rightarrow[0,1]\) and \(0\leq\mu_{A}(x)+\
u_{A}(x)\leq 1\) for each \(x\in X\). Meanwhile, a degree of hesitation (intuitionistic fuzzy index) of an element \(x\) in IFS \(A_{I}\) is defined as \(\pi_{A_{I}}(x)=1-\mu_{A}(x)-\
u_{A}(x)\). A method to construct an IFS from a fuzzy set is given in Proposition 1, Theorem 1, and Corollary 1, which are cited from [34; 35]. Proposition 1 ([34; 35]): _Let \(F\) be a mapping that is defined as \(F:[0,1]^{2}\times[0,1]\to L^{*}\) with \(L^{*}=\{(x,y)|(x,y)\in[0,1]\times[0,1]\ \ \&\ \ x+y\leq 1\}\), and \(F(x,y,\delta)=(F_{\mu}(x,y,\delta),F_{v}(x,y,\delta))\), where \(F_{\mu}(x,y,\delta)=x(1-\delta y)\) and \(F_{v}(x,y,\delta)=1-x(1-\delta y)-\delta y\). Let \(\pi^{*}:L^{*}\rightarrow[0,1]\) be a function defined as \(\pi^{*}(u,v)=1-u-v\) for \(u,v\in[0,1]\). The mapping \(F\) satisfies the following conditions:_  
1. _If_ \(y_{1}\leq y_{2}\)_, then_ \(\pi^{*}(F(x,y_{1},\delta))\leq\pi^{*}(F(x,y_{2},\delta))\) _for_ \(x\in[0,1]\)_,_
2. \(F_{\mu}(x,y,\delta)\leq x\leq 1-F_{v}(x,y,\delta)\) _for_ \(x\in[0,1]\)_,_
3. \(F(x,0,\delta)=(x,1-x)\)_,_
4. \(F(0,y,\delta)=(0,1-\delta y)\)_,_
5. \(F(x,y,0)=(x,1-x)\)_,_
6. \(\pi^{*}(F(x,y,\delta))=\delta y\)_._  
Theorem 1 ([34; 35]): _Let \(FS(X)\) be a set of all fuzzy set in \(X\) and \(A\subseteq X\). Let \(A_{F}=\{(x,\mu_{A}(x))|\ x\in X\}\) be a fuzzy set in \(FS(X)\), where \(\mu_{A}:X\rightarrow[0,1]\). Let \(\pi_{\cdot},\xi:X\rightarrow[0,1]\) be two functions defined on \(X\). The set_  
\[A_{I}=\{(x,F(\mu_{A}(x),\pi(x),\xi(x))|x\in X\}\]  
_is an Atanassov IFS, where the function \(F\) is defined as in Proposition 1._  
Corollary 1 ([34]): _Let \(\pi^{*}\) and \(F\) be two functions as defined in Proposition 1. If we choose \(\xi(x)=1\) for each \(x\in X\) in Theorem 1, then_  
\[\pi(x)=\pi^{*}(F(\mu_{A}(x),\pi(x),1)).\]  
_Under the condition in Corollary 1, we obtain an IFS:_  
\[A_{I}=\{(x,\mu_{A}(x)(1-\pi(x)),1-(\mu_{A}(x)(1-\pi(x)))-\pi(x)|x\in X\}. \tag{1}\]  
The non-membership degree of IFS \(A_{I}\) in (1) will be used in the implementation of coloring of PFGs in Section 5. Furthermore, the notion of a picture fuzzy set (PFS) and the construction of a PFS from an IFS are discussed. Definition 1 ([8]): Given a universal set \(X\) and \(A\subseteq X\). A set of the form \(\tilde{A}=\{(v,\mu_{A}(v),\eta_{A}(v),\)\(\
u_{A}(v))|v\in X\}\) is mentioned as a PFS on \(X\), wherein \(\mu_{A}(v)\in[0,1]\) is a membership degree that describes the truth value of existence of element \(v\) in \(\tilde{A}\), \(\eta_{A}(v)\in[0,1]\) is a NeuM degree that represents the indeterminacy degree of existence of \(v\) in \(\tilde{A}\), and \(\
u_{A}(v)\in[0,1]\) is a non-membership degree that shows the falsity degree of existence of \(v\) in PFS \(\tilde{A}\), such that \(0\leq\mu_{A}(v)+\eta_{A}(v)+\
u_{A}(v)\leq 1\). The value \(\pi_{A}(v)=1-(\mu_{A}(v)+\eta_{A}(v)+\
u_{A}(v))\) is called a refusal degree of membership of \(v\) in \(\tilde{A}\). A method to construct a PFS from an IFS is given in Theorem 2.