## 3 Preliminaries
Theorem 2 ([36]): _If \(A_{I}=\{(x,\mu_{A}(x),\
u_{A}(x))|x\in X\}\) is an IFS on \(X\) and \(g:[0,1]\rightarrow[0,1]\) is any function such that \(g(0)=0\) and \(g(x)\leq x\), then_  
\[A_{p}=\{(x,P_{p}(\mu_{A}(x),\
u_{A}(x)),P_{\eta}(\mu_{A}(x),\
u_{A}(x)),P_{v}( \mu_{A}(x),\
u_{A}(x)))\}\]  
_is a PFS on \(X\) wherein the mapping \(P:[0,1]\times[0,1]\rightarrow[0,1]^{2}\times[0,1]\) is defined by \(P(u,v)=(P_{\mu}(u,v),P_{\eta}(u,v),P_{v}(u,v))\) with \(P_{\mu}(u,v)=u,P_{\eta}(u,v)=g(1-u-v)\), and \(P_{v}(u,v)=v\)._  
In other words, \(P_{\mu}(\mu_{A}(x),\
u_{A}(x))=\mu_{A}(x)\), \(P_{\eta}(\mu_{A}(x),\
u_{A}(x))=g(1-\mu_{A}(x)-\
u_{A}(x))\), and \(P_{v}(\mu_{A}(x),\
u_{A}(x))=\
u_{A}(x)\) for \(x\in X\). Further, the function \(g\) in Theorem 2 is called a neutral or refusal membership function of PFS \(A_{p}\), and it will be used in Section 5. In Definitions 2 and 3, we present ideas of an empty PFS and a universal PFS, which are cited from [37]. Definition 2 ([37]): Let \(\tilde{A}=\{(v,\mu_{A}(v),\eta_{A}(v),\
u_{A}(v))\}\) be a PFS on \(X\) and \(A\subseteq X\). The set \(\tilde{A}\) is called an empty PFS if \(\mu_{A}(v)=0,\eta_{A}(v)=0\), and \(\
u_{A}(v)=1\) for each \(v\in X\). The empty PFS is denoted by \(\mathcal{O}_{pfs}\). Definition 3 ([37]): Given PFS \(\tilde{A}\) in Definition 2, the set \(\tilde{A}\) is named a universal PFS if \(\mu_{A}(v)=1\), \(\eta_{A}(v)=0\), and \(\
u_{A}(v)=0\) for each \(v\in X\). Additionally, we provide information on the picture fuzzy subset in Definition 4, which is quoted from [8]. Definition 4 ([8]): Let \(X\) be a universal set and \(A,B\subseteq X\). Given two PFSs on \(X\): \(\tilde{A}=\{(a,\mu_{A}(a),\)\(\eta_{A}(a),\
u_{A}(a))\}\) and \(\tilde{B}=\{(b,\mu_{B}(b),\eta_{B}(b),\
u_{B}(b))\}\), \(a,b\in X\). The PFS \(\tilde{A}\) is mentioned as the picture fuzzy subset of \(\tilde{B}\), denoted by \(\tilde{A}\subseteq\tilde{B}\), if  
\[\mu_{A}(v)\leq\mu_{B}(v),\eta_{A}(v)\leq\eta_{B}(v),\
u_{A}(v)\geq\
u_{B}(v)\]  
for all \(v\in X\). The notion of PFS is used as a basis to define a PFG, as described in Definition 5. Definition 5 ([9]): We assume that \(X\) is a universal set that contains vertices. We mention a graph \(\tilde{G}=(\tilde{V},\tilde{E})\) as a PFG if \(\tilde{V}=\{(x,\mu_{1}(x),\eta_{1}(x),\
u_{1}(x))\}\) is a picture fuzzy vertex set (PFVS) on \(X\) with the membership, neutral, and non-membership functions as follows: \(\mu_{1},\eta_{1},\
u_{1}:X\rightarrow[0,1]\), in which \(0\leq\mu_{1}(x)+\eta_{1}(x)+\
u_{1}(x)\leq 1\) for each \(x\in X\). Meanwhile, \(\tilde{E}=\{(xy,\mu_{2}(xy),\eta_{2}(xy),\
u_{2}(xy))\}\) is a picture fuzzy edge set (PFES) on \(E\subseteq X\times X\) with the mem bership, neutral membership, and non-membership functions as follows:_  
\(\mu_{2},\eta_{2},\
u_{2}:X\times X\rightarrow[0,1]\)_, such that_  
\[\mu_{2}(xy)\leq\min\{\mu_{1}(x),\mu_{1}(y)\},\eta_{2}(xy)\leq\min\{\eta_{1}(x), \eta_{1}(y)\},\
u_{2}(xy)\leq\max\{\
u_{1}(x),\
u_{1}(y)\},\]  
_and \(0\leq\mu_{2}(xy)+\eta_{2}(xy)+\
u_{2}(xy)\leq 1\) for each \(xy\in E\)._  
_The value of \(\mu_{1}(x)\) is a membership degree that represents the truth value of existence of vertex \(x\) in \(X\), \(\eta_{1}(x)\) is NeuM degree that describes the indeterminacy degree of existence of \(x\) in \(X\), and \(\
u_{1}(x)\) is non-membership degree that shows the falsity degree of the existence of vertex \(x\) in \(X\). Meanwhile, the values of \(\mu_{2}(xy),\eta_{2}(xy),\
u_{2}(xy)\) represent a membership degree that tells the truth of adjacency of \(x\) and \(y\), an NeuM degree that describes the indeterminacy of adjacency of \(x\) and \(y\), and a non-membership degree that shows the falsity of adjacency between \(x\) and \(y\), respectively._  
Furthermore, we discuss the concepts of underlying graph, picture fuzzy subgraph, and a complete picture fuzzy graph (CPFG) that will be used in the next section.