Normal m -domination and inverse m -domination in Pythagorean fuzzy graphs with application in decision making

Abstract

In this paper, the notions of normal m-dominating set, normal m-domination number, inverse normal domination set (number) and inverse normal m-domination number are introduced, and some the related results are investigated. Finally, a utilization relevant to decision-making based on influencing factors the company’s efficiency is presented.

Keywords

Pythagorean fuzzy graph inverse normal domination set (number)

1 Introduction

The idea of graph theory was firstly offered by Euler in 1736. In the history of mathematics, Euler’s offered solution to the famous Koenigsberg bridge issues is considered the first theorem of graph theory. Graph theory is set up as a very beneficial tool for solving hybrid problems in diverse fields such as geometry, algebra, number theory, topology, operations research, optimization, and computer science. Cockayne and Hedetniemi [9] offered the domination parameter and the independence parameter. As well as, Zadeh [27] first offered the idea of fuzzy sets. Then, Rosenfeld [22] offered the notion of fuzzy graph theory as an extension of Euler graph, and many researchers have followed the study in this field [1 , 23]. Further, Janakiram and Kulli [13] is offered the notion of the cobondage number in graphs. In other perusal, Somasundram [24] is offered the notion of domination in fuzzy graphs. The notion of domination in intuitionists fuzzy graphs is offered by Parvathi and Thamizhendhi [21]. As well, Nagoor Gani and Prasanna Devi [19] offered the diminution in the domination number of a fuzzy graph. In some practical problems, fuzzy sets are perhaps not sufficient to represent the obscurity of the human mind. Yager [25, 26] introduced the notion of the Pythagorean fuzzy set (PFS) as an extension of the intuitionists fuzzy set (IFS) to administer the complex impreciseness and uncertainty in practical decision-making issues. The outstanding characteristic sum of its membership degree and non-membership degree is no more significant than one, with the square sum of its membership degree and non-membership degree no more significant than one. In addition, Naz [20] defined the notion of the Pythagorean fuzzy graph. The notions of dominating sets and cobondage sets are considered as the fundamental notions in the theory of fuzzy graphs, and have usages in several fields, particularly in the fields of operations research, neural networks, electrical networks, and monitoring communication. Banitalebi and Borzooei [7] offered the notion of normal dominating set in the Pythagorean fuzzy graph.

The purpose of this paper is to discuss the notions of m-dominating set, normal m-domination number, inverse normal domination set (number), and inverse normal m-domination number in Pythagorean fuzzy graph, and finally, a model for optimizing the normal domination number parameter will be presented, in which it will be possible to optimize the normal domination number parameter more accurately in a partial way.

2 Preliminaries

A fuzzy graph G = (σ, ψ) on crisp graph $G^{*} = (𝕍, 𝔼)$ is a pair of mappings $σ : 𝕍 \to [0, 1]$ and $ψ : 𝔼 \to [0, 1]$ where, for any $zw \in 𝔼,$ ψ (zw) ⪯ min {σ (z) , σ (w)}.

A Pythagorean fuzzy set A on $𝕏$ is a pair (ψ_A, φ_A), where the mappings ψ_A : X → [0, 1] and φ_A : X → [0, 1] describes the degree of membership and the degree of nonmembership of the element $t \in 𝕏$ to the set A, respectively, where for all $t \in 𝕏$ , 0 ⪯ (ψ_A (t)) ² + (φ_A (t)) ² ⪯ 1. $I_{A} (t) = \sqrt{1} - {(ψ_{A} (t))}^{2} - {(ϕ_{A} (t))}^{2}$ is named the Pythagorean fuzzy index of element $t \in 𝕏$ (see [1]).

A Pythagorean fuzzy graph(PyFG) on simple graph $G^{*} = (𝕍, 𝔼)$ is interpreted to be a pair $G = (ℙ, ℚ)$ , where $ℙ = (ψ_{ℙ}, φ_{ℙ})$ is a Pythagorean fuzzy set on $𝕍$ and $ℚ = (ψ_{ℚ}, φ_{ℚ})$ is a Pythagorean fuzzy set on $𝔼$ where for each arc $zw \in 𝔼$ , $ψ_{ℚ} (zw) ⪯ ψ_{ℙ} (z) \land ψ_{ℙ} (w), φ_{ℚ} (zw) ⪯ φ_{ℙ} (z) \lor φ_{ℙ} (w) .$

Definition 2.1. [3] Put $G = (ℙ, ℚ)$ be a PyFG on simple graph $G^{*} = (𝕍, 𝔼)$ . Then:

(i) G is named a complete PyFG if for each $r, s \in 𝕍,$ $ψ_{ℚ} (rs) = ψ_{ℙ} (r) \land ψ_{ℙ} (s), φ_{ℚ} (rs) = φ_{ℙ} (r) \lor φ_{ℙ} (s) .$ (ii) G is named a strong PyFG if for each $rs \in 𝔼,$ $ψ_{ℚ} (rs) = ψ_{ℙ} (r) \land ψ_{ℙ} (s)}, φ_{ℚ} (rs) = φ_{ℙ} (r) \lor φ_{ℙ} (s) .$

Definition 2.2. [7] Put G be a PyFG. Then,

(i) the order of G is interpreted with, $| 𝕍 | = \sum_{s_{i} \in 𝕍} (\frac{ψ_{ℙ} (s_{i}) + \sqrt{1 - (φ_{ℙ} (s_{i}))^{2}}}{2}) .$ (ii) The size of G is interpreted with, $| 𝔼 | = \sum_{r_{i} r_{j} \in 𝔼} (\frac{ψ_{ℚ} (r_{i} r_{j}) + \sqrt{1 - (φ_{ℚ} (r_{i} r_{j}))^{2}}}{2}) .$ (iii) The cardinality of G is interpreted with, $| G | = | 𝕍 | + | 𝔼 | .$ (iv) For any $𝕊 \subseteq 𝕍$ , the node cardinality of $𝕊$ is indicated with, $O (𝕊)$ and interpreted with, $O (𝕊) = \sum_{s_{i} \in 𝕊} (\frac{ψ_{ℙ} (s_{i}) + \sqrt{1 - (φ_{ℙ} (s_{i}))^{2}}}{2}) .$ (v) For any $ℝ \subseteq 𝔼$ , the arc cardinality of $ℝ$ is indicated with $S (ℝ)$ and interpreted with, $S (ℝ) = \sum_{r_{i} r_{j} \in ℝ} (\frac{ψ_{ℚ} (r_{i} r_{j}) + \sqrt{1 - (φ_{ℚ} (r_{i} r_{j}))^{2}}}{2}) .$

Definition 2.3. [7] An arc $rs \in 𝔼$ is named a strong arc if $ψ_{ℚ} (rs) ⪯ ψ_{ℚ}^{\infty} (rs) and φ_{ℚ} (rs) ⪯ φ_{ℚ}^{\infty} (rs) .$

Definition 2.4. [7] The neighborhood of $r \in 𝕍$ is indicated with N (r) and interpreted as follows: $N (r) = {s \in V ∣ rs is a strong arc in G} .$

Definition 2.5. [7] An arc e in G is named a normal arc if $(ψ_{ℚ} (e))^{2} + (φ_{ℚ} (e))^{2} = 1,$ and so $ψ_{ℚ} (e) = S ({e}) .$

Definition 2.6. [7] The normal neighborhood of $r \in 𝕍$ is indicated with N (r) and interpreted as follows: $N_{n} (r) = {s \in 𝕍 ∣ rs is a normal arc in G} .$

Definition 2.7 [7] Put G be a PyFG. Then:

(i) r normally dominates s in G if there is a normal arc betwixt r and s .

(ii) $𝕌 \subset 𝕍$ is named a normal dominating set (NDS) in G if for any $s \in 𝕍 \ 𝕌$ , there is $r \in 𝕌$ where r normally dominates s .

(iii) A NDS $𝕌$ in G is named a minimal NDS if no proper subset of U is a NDS.

(iv) Minimum node cardinality amid all minimal NDSs of G is named a lower normal domination number of G and indicated with $(nd)_{𝕍} (G)$ .

(v) Maximum node cardinality amid all minimal NDSs of G is named an upper normal domination number of G and indicated with $(ND)_{𝕍} (G)$ .

(vi) The normal domination number of G is indicated with $(N Δ)_{𝕍} (G)$ and interpreted as follows, $(N Δ)_{𝕍} (G) = \frac{(nd)_{𝕍} (G) + (ND)_{𝕍} (G)}{2} .$

Definition 2.8. [7] Presume G be a PyFG. Then,

(i) two nodes $r, s \in 𝕍$ are named abnormally independent if there is not any normal arc betwixt them,

(ii) $𝕌 \subset 𝕍$ is named an abnormal independent set (AIS) in G if for any $r, s \in 𝕌$ , $(ψ_{ℚ} (rs))^{2} + (φ_{ℚ} (rs))^{2} < 1,$ (iii) an AIS $𝕌$ in G is named a maximal AIS if for any node $s \in 𝕍 \ 𝕌$ , the set $𝕌 \cup {s}$ is not AIS,

(iv) minimum node cardinality amid all maximal AISs is named a lower abnormal independent number of G and indicated with $(ai)_{𝕍} (G),$

(v) maximum node cardinality amid all maximal AISs is named an upper abnormal independent number of G and indicated with $(AI)_{𝕍} (G) .$

Definition 2.9. [7] The independent number of G is indicated with AI (G) and interpreted as follows, $AI (G) = \frac{(ai)_{𝕍} (G) + (AI)_{𝕍} (G)}{2} .$

Notation. Henceforth, in this article we presume $G = (ℙ, ℚ)$ be a PyFG on crisp graph $G^{*} = (𝕍, 𝔼)$ and indicated with PyFG.

3 Normal m-dominating set and inverse normal m-dominating set in PyFGs

In this part, we described the notions of normal m-dominating set and inverse normal m-dominating set in PyFGs and investigate some related results.

Definition 3.1. $D_{n}^{m} \subset 𝕍$ is named a normal m-dominating set (Nm-DS) of G if for any node $r \in 𝕍 \ D_{n}^{m}$ we get r normally dominate with at least m nodes in $D_{n}^{m}$ .

Definition 3.2. (i) A Nm-DS $D_{n}^{m}$ of G is named a minimal Nm-DS if no proper subset of $D_{n}^{m}$ is a Nm-DS of G.

(ii) Minimum node cardinality amid all minimal Nm-DSs of G is named lower normal m-domination number of G and is indicated with $(nd)_{𝕍}^{m} (G) .$

(iii) Maximum node cardinality amid all minimal Nm-DSs of G is named normal upper m-domination number of G and is indicated with $(ND)_{𝕍}^{m} (G) .$

(iv) The normal m-domination number of G is indicated with $(N Δ)_{V}^{m} (G)$ and interpreted as follows as: $(N Δ)_{𝕍}^{m} (G) = \frac{(nd)_{𝕍}^{m} (G) + (ND)_{𝕍}^{m} (G)}{2} .$

Example 3.3. Put the PyFG G as Fig. 1.

Fig. 1

PyFG G.

Obviously, D = {b, d} is a minimal N2DS in G. By routine calculation, $(nd)_{𝕍}^{2} (G) = (ND)_{𝕍}^{2} (G) = (N Δ)_{𝕍}^{2} (G) = 1.54 .$

Theorem 3.4. Put G be a PyFG and $D_{n}^{m}$ be an abnormal independent and Nm-DS of G. Then $D_{n}^{m}$ is a minimal NDS of G.

Proof. If $D_{n}^{m}$ is an abnormal independent and Nm-DS of PyFG G, then $D_{n}^{m}$ is a NDS of G and $D_{n}^{m} - {z}$ is not a NDS for each $z \in D_{n}^{m}$ . Hence $D_{n}^{m}$ is a minimal NDS of G. □

Theorem 3.5. A Nm-DS $D_{n}^{m}$ of a PyFG G is a minimal Nm-DS iff for any node $r \in D_{n}^{m}$ , one of the following authentic qualifications.

(i) $| N_{n} (r) \cap D_{n}^{m} | \leq m - 1$ ,

(ii) there are nodes $s \in 𝕍 \ D_{n}^{m} and w_{1}, w_{2}, w_{3}, . . . w_{m - 1} \in D_{n}^{m}$ such that $N_{n} (s) \cap (D_{n}^{m} - {r}) = {w_{1}, w_{2}, w_{3}, . . . w_{m - 1}} .$

Proof. Put $D_{n}^{m}$ is a minimal Nm-DS of G. Then, for any node $r \in D_{n}^{m}$ , $D_{n}^{m} - {r}$ is not a Nm-DS. Thus there exists $s \in 𝕍 \ (D_{n}^{m} - {r})$ , which is not normally dominated by m nodes in $D_{n}^{m} - {r}$ . If s = r, $| N_{n} (r) \cap D_{n}^{m} | \leq m - 1$ . If s ≠ r, then s is not normally dominated with m nodes in $D_{n}^{m} - {r}$ , but is normally dominated by m nodes in $D_{n}^{m}$ . Hence there are nodes $w_{1}, w_{2}, w_{3}, . . . w_{m - 1} \in D_{n}^{m}$ such that $N_{n} (s) \cap (D_{n}^{m} - {r}) = {w_{1}, w_{2}, w_{3}, . . . w_{m - 1}}$ .

Conversely, consider that $D_{n}^{m}$ is not a minimal Nm-DS. Then there exists a node $r \in D_{n}^{m}$ where $D_{n}^{m} - {r}$ is a Nm-DS. Thus r is a normal neighbor of at least m nodes in $D_{n}^{m} - {r}$ , and also (i) does not true. As well as, any node in $𝕍 \ D_{n}^{m}$ is a normal neighbor of at least m nodes in $D_{n}^{m} - {r}$ , and so (ii) does not authentic, which is a contradiction. Since at least one of the conditions should be true. Therefore, $D_{n}^{m}$ is a minimal Nm-DS. □

Proposition 3.6. (i) If |N_n (z) | ≤ m - 1, for $z \in 𝕍$ , then z belongs to the every Nm-DSs of G.

(ii) $(nd)_{𝕍}^{m} (G) \geq (nd)_{𝕍} (G)$ and $(ND)_{𝕍}^{m} (G) \geq (ND)_{𝕍} (G)$ and so $(N Δ)_{𝕍}^{m} (G) \geq (N Δ)_{𝕍} (G) .$

Proof. The proof is straightforward. □

Theorem 3.7. Put M be a minimal Nm-DS (m > 1). Then $𝕍 \ M$ is not necessarily Nm-DS of G.

Proof. Assume $z \in 𝕍$ and |N_n (z) | ≤ m - 1. Then, z belongs to every Nm-DS of G. Thus z ∈ M, and so z do not belong to $𝕍 \ M$ . Therefore, $𝕍 \ M$ is not a Nm-DS of G. □

Note. If D_n is a minimal NDS of a PyFG $G = (ℙ, ℚ)$ without abnormal isolated node, then $𝕍 \ D_{n}$ is a NDS of G.

Definition 3.8. Let $D_{n} \subset 𝕍$ be a minimal NDS of G. Then:

(i) $D_{n}^{- 1} \subseteq 𝕍 \ D_{n}$ is named an inverse NDS of G concerning D_n if $D_{n}^{- 1}$ is a NDS of G.

(ii) An inverse NDS $D_{n}^{- 1} \subseteq 𝕍 \ D_{n}$ is named a minimal inverse NDS of G concerning D_n if no proper subset of $D_{n}^{- 1}$ is a NDS.

(iii) The lower inverse normal domination number of G is indicated with $(nd)_{𝕍}^{- 1} (G)$ and interpreted with, $\begin{matrix} (nd)_{𝕍}^{- 1} (G) \\ = min {O (D_{n}^{- 1}) | D_{n}^{- 1} a minimal inverse NDS of G} . \end{matrix}$ (iv) The upper inverse normal domination number of G is indicated with $(ND)_{𝕍}^{- 1} (G)$ and interpreted with, $\begin{matrix} (ND)_{𝕍}^{- 1} (G) \\ = max {O (D_{n}^{- 1}) | D_{n}^{- 1} a minimal inverse NDS of G} . \end{matrix}$ (v) The inverse normal domination number of G is indicated with $(N Δ)_{𝕍}^{- 1} (G)$ and interpreted with, $(N Δ)_{𝕍}^{- 1} (G) = \frac{(nd)_{𝕍}^{- 1} (G) + (ND)_{𝕍}^{- 1} (G)}{2} .$

Example 3.9. Put the PyFG G as Fig. 2.

Fig. 2

PyFG G.

Clearly, D = {u₃, u₅, u₈} is a minimal NDS of G. Also, $𝕍 \ D_{n} = {u_{1}, u_{2}, u_{4}, u_{6}, u_{7}}$ is an AIS and NDS of G. Hence, $𝕍 \ D_{n}$ is a minimal NDS. Therefore, $𝕍 \ D_{n}$ a minimal inverse NDS with respect to D_n of G. By routine calculations, $\begin{matrix} (nd)_{𝕍}^{- 1} (G) = (nd)_{𝕍} (G) = 1.97, \\ (ND)_{𝕍}^{- 1} (G) = (ND)_{𝕍} (G) = 3.48, \end{matrix}$ and so $(N Δ)_{𝕍}^{- 1} (G) = (N Δ)_{𝕍} (G) = 2.725 .$

Theorem 3.10. A minimal NDS D_n of G is a minimal inverse NDS if G does not have any abnormal isolated node.

Proof. Assume that D_n is a minimal inverse NDS and G has an abnormal isolated node $s \in 𝕍$ , with the contrary. Then s belongs to every NDS D_n of G. Thus, s does not belong to $𝕍 \ D_{n}$ . Hence, $𝕍 \ D_{n}$ is not contain NDS of G, which is a contradiction. Therefore, G does not have any abnormal isolated node.

Conversely, consider that D_n is a minimal NDS of G, and G does not have any abnormal isolated node. Then any node in D_n is normally dominated with at least one node in $𝕍 \ D_{n}$ . Hence $𝕍 \ D_{n}$ is a NDS, and so $𝕍 \ D_{n}$ contains a minimal NDS of G. Therefore, D_n is a minimal inverse NDS of G concerning $D_{n}^{- 1}$ . □

Theorem 3.11. If G has an abnormal isolated node $z \in 𝕍$ , then $(N Δ)_{𝕍}^{- 1} (G) = 0 .$

Proof. Assume that $z \in 𝕍$ where |N_n (z) |=0 . Then, z belongs to every NDS D_n of G. Thus z does not belong to $𝕍 \ D_{n}$ . Hence, $𝕍 \ D_{n}$ does not contain any NDS of G. Therefore, $(N Δ)_{𝕍}^{- 1} (G) = 0 .$ □

Note. For each PyFG G without an abnormal isolated node, we have $(nd)_{𝕍} (G) \leq (nd)_{𝕍}^{- 1} (G) .$

Theorem 3.12. If G is a PyFG and D_n is a minimal NDS of G, then $O (D_{n}) + O (D_{n}^{- 1}) \leq | 𝕍 | .$ Furthermore, equality holds if G does not have any abnormal isolated node, and $𝕍 \ D_{n}$ is an AIS.

Proof. Put G be a PyFG, D_n be a minimal NDS and $D_{n}^{- 1}$ be a minimal inverse NDS concerning D_n of G. Then $D_{n}^{- 1} \subseteq 𝕍 \ D_{n}$ . Thus $O (D_{n}^{- 1}) \leq O (𝕍 \ D_{n}),$ and so $O (D_{n}^{- 1}) \leq | 𝕍 | - O (D_{n}) .$ Hence, $O (D_{n}^{- 1}) + O (D_{n}) \leq | 𝕍 | .$ If G does not have any abnormal isolated node and D_n is a minimal NDS, then $𝕍 \ D_{n}$ is a NDS and so $𝕍 \ D_{n}$ is an inverse NDS to D_n. Since $𝕍 \ D_{n}$ is an AIS in G, we obtain $𝕍 \ D_{n}$ is a minimal inverse NDS of G for D_n. □

Corollary 3.13. If G has an inverse NDS, then $(nd)_{𝕍} (G) + (nd)_{𝕍}^{- 1} (G) \leq | 𝕍 | .$

Note 3.14. If $D_{n} \subset 𝕍$ is a minimal NDS of G, then NDS $D_{n}^{- 1} \subseteq 𝕍 \ D_{n}$ is a minimal inverse NDS of G with respect to D_n iff for any node $r \in D_{n}^{- 1}$ , one of the following authentic qualifications.

(i) $N_{n} (r) \subseteq 𝕍 \ D_{n}^{- 1},$

(ii) There is a node $s \in 𝕍 \ D_{n}^{- 1}$ where $N_{n} (s) \cap D_{n}^{- 1} = {r} .$

Theorem 3.15. If $(nd)_{𝕍} (G) > \frac{| 𝕍 |}{2},$ then G has no inverse NDS.

Proof. Put D_n is a minimal NDS of G and $O (D_{n}) = (nd)_{𝕍} (G) > \frac{| 𝕍 |}{2} .$ If $D_{n}^{- 1} \subseteq 𝕍 \ D_{n}$ is a minimal inverse NDS to D_n and $(nd)_{𝕍}^{- 1} (G) \leq O (D_{n}^{- 1}),$ with the contrary, then with Theorem 3, $O (D_{n}) + O (D_{n}^{- 1}) \leq | 𝕍 |,$ and so, $O (D_{n}^{- 1}) \leq | 𝕍 | - O (D_{n}),$ thus, $O (D_{n}^{- 1}) < \frac{| 𝕍 |}{2},$ hence, $(nd)_{𝕍}^{- 1} (G) < (nd)_{𝕍} (G),$ which is a contradiction. Therefore, G does not have any inverse NDS. □

Corollary 3.16. If G has an inverse NDS, then $(nd)_{𝕍} (G) \leq \frac{| 𝕍 |}{2} .$

Notation. If in a graph $G^{*} = (𝕍, 𝔼)$ we add an arc e to E, then we indicate it with $𝔼_{e} = 𝔼 \cup {e}$ and $G_{e}^{*} = (𝕍, 𝔼_{e})$ . Moreover, if a PyFG $G = (ℙ, ℚ)$ on G^* extends on $G_{e}^{*}$ , then we indicate it with $G_{e} = (ℙ_{e}, ℚ_{e})$ .

Notation 3.17. If arc e in PyFG G_e is a normal arc, then we indicate $G_{e}^{n} = (ℙ_{e}^{n}, ℚ_{e}^{n})$ in lieu of $G_{e} = (ℙ_{e}, ℚ_{e})$ .

Theorem 3.18. Put G does not have any abnormal isolated node and e = rs is an additional normal arc in $G_{e}^{*} .$ Then, $(N Δ)_{𝕍}^{- 1} (G_{e}^{n}) \leq (N Δ)_{𝕍}^{- 1} (G) .$

Proof. If S is a minimal inverse NDS of G and r, s ∈ S where $N_{n} (r) \subseteq 𝕍 \ S,$ then with adding e = rs, S \ {r} is a minimal inverse NDS in $G_{e}^{n} .$ Otherwise, S is a minimal inverse NDS in $G_{e}^{n} .$ Therefore, $(N Δ)_{𝕍}^{- 1} (G_{e}^{n}) \leq (N Δ)_{𝕍}^{- 1} (G) .$ □

Theorem 3.19. For any PyFG G, $(nd)_{𝕍}^{- 1} (G) \leq (ai)_{𝕍} (G) .$

Proof. We investigate the following cases:

Case 1: If G has an abnormal isolated node, then the proof is clear and $(nd)_{𝕍}^{- 1} (G) \leq (ai)_{𝕍} (G) .$

Case 2: If G does not have any abnormal isolated node and S is a maximal AIS of G, where $O (S) = (ai)_{𝕍} (G),$ then S is a minimal NDS. Also $𝕍 \ S$ is a NDS, and so $𝕍 \ S$ contains a minimal NDS D_n of G. Hence S is a minimal inverse NDS of G to D_n. Therefore $(nd)_{𝕍}^{- 1} (G) \leq (ai)_{𝕍} (G) .$ □

Corollary 3.20. For any PyFG G, $(nd)_{𝕍} (G) \leq (nd)_{𝕍}^{- 1} (G) \leq (ai)_{𝕍} (G) .$

Theorem 3.21. For any PyFG G without an abnormal isolated node, $(nd)_{𝕍} (G) \leq \frac{| 𝕍 | + (nd)_{𝕍}^{- 1} (G)}{3} .$

Proof. Assume that $G = (ℙ, ℚ)$ be a PyFG. Then with Theorem 3, $(nd)_{𝕍} (G) \leq \frac{| 𝕍 |}{2},$ also $(nd)_{𝕍} (G) \leq (nd)_{𝕍}^{- 1} (G) .$ Hence the result holds. □

Definition 3.22. Let $D_{n}^{m} \subset 𝕍$ is a minimal Nm-DS of G. Then:

(i) $D_{n}^{- m} \subseteq 𝕍 \ D_{n}^{m}$ has named an inverse Nm-DS of G concerning $D_{n}^{m}$ if $D_{n}^{- m}$ is a Nm-DS of G.

(ii) An inverse Nm-DS $D_{n}^{- m} \subseteq 𝕍 \ D_{n}^{m}$ is named a minimal inverse Nm-DS of G concerning $D_{n}^{m}$ if no proper subset of $D_{n}^{- m}$ is a Nm-DS.

(iii) The lower inverse normal m-domination number of G is indicated with $(nd)_{𝕍}^{- m} (G)$ and interpreted with, ${(n d)}_{V}^{- m} (G) = \min {O (D_{n}^{- m}) | D_{n}^{- m} is a minimal inverse Nm-DS of G} .$ (iv) The upper inverse normal m-domination number of G is indicated with $(ND)_{𝕍}^{- m} (G)$ and interpreted with, $\begin{matrix} (ND)_{𝕍}^{- m} (G) = \\ max {O (D_{n}^{- m}) | D_{n}^{- m} Nm - DS of G} . \end{matrix}$ (v) The inverse m-domination number of G is indicated with $(N Δ)_{𝕍}^{- m} (G)$ and interpreted with, $\begin{matrix} (N Δ)_{𝕍}^{- m} (G) = \frac{(nd)_{𝕍}^{- m} (G) + (ND)_{𝕍}^{- m} (G)}{2} . \end{matrix}$

Theorem 3.23. A minimal Nm-DS $D_{n}^{m}$ of G is a minimal inverse Nm-DS if G does not have any node $z \in 𝕍$ where |N_n (z) | ≤ m - 1 .

Proof. Assume that $D_{n}^{m}$ is a minimal inverse Nm-DS and G has a node $z \in 𝕍$ such that |N_n (z) | ≤ m - 1, with the contrary. Then z belongs to every Nm-DS of G. Thus $z \in D_{n}^{m}$ and so u does not belong to $𝕍 \ D_{n}^{m}$ . Hence, $𝕍 \ D_{n}^{m}$ does not contain Nm-DS of G, which is a contradiction. Therefore, G does not have any node $z \in 𝕍$ such that |N_n (z) | ≤ m - 1 . □

Theorem 3.24. If G has a node $z \in 𝕍$ where |N_n (u) | ≤ m - 1, then $(N Δ)_{𝕍}^{- m} (G) = 0 .$

Proof. Assume that $z \in 𝕍$ where |N_n (u) | ≤ m - 1 . Then z belongs to every Nm-DS of G. Thus $z \in D_{n}^{m}$ , and so z does not belong to $𝕍 \ D_{n}^{m}$ . Hence, $𝕍 \ D_{n}^{m}$ does not contain in any Nm-DS of G. Therefore, $(N Δ)_{𝕍}^{- m} (G) = 0 .$ □

Theorem 3.25. For each PyFG G, a node $s \in 𝕍 \ D_{n}^{k}$ , belongs to every inverse Nk-DS $D_{n}^{- k}$ of G concerning $D_{n}^{k}$ if k ≤ |N_n (s) | ≤ k + 1 .

Proof. Assume that there is node $s \in 𝕍 \ D_{n}^{k}$ so that k ≤ |N_n (s) | ≤ k + 1 . Then s has exactly k or k + 1 normal neighbors in V and $k \leq | N_{n} (s) \cap D_{n}^{k} | \leq k + 1 .$ Put $D_{n}^{- k} \subseteq 𝕍 \ D_{n}^{k}$ be an inverse Nk-DS of G concerning $D_{n}^{k} .$ Hence, with Theorem 3, $s \in D_{n}^{- k} .$ Therefore, s belongs to every inverse Nk-DS $D_{n}^{- k}$ of G concerning $D_{n}^{k} .$ □

Note 3.26. (i) If G does not have any node $z \in 𝕍$ where |N_n (z) | ≤ m - 1, we have $(nd)_{𝕍}^{m} (G) \leq (nd)_{𝕍}^{- m} (G) .$ (ii) If G is a PyFG and $D_{n}^{m}$ is a minimal Nm-DS of G, then $O (D_{n}^{m}) + O (D_{n}^{- m}) \leq | 𝕍 | .$ (iii) If G has an inverse Nm-DS, then $(nd)_{𝕍}^{m} (G) + (nd)_{𝕍}^{- m} (G) \leq | 𝕍 | .$ (iv) If $(nd)_{𝕍}^{m} (G) > \frac{| 𝕍 |}{2},$ then G has no inverse Nm-DS.

Theorem 3.27. G has at least two distinct Nm-DSs iff $(N Δ)_{𝕍}^{- m} (G) > 0 .$

Proof. Assume that D₁ and D₂ are two distinct Nm-DSs of G. Then D₁ and D₂ contain minimal Nm-DSs $D_{1}^{'}$ and $D_{2}^{'},$ respectively. Hence, $D_{1}^{'} \subseteq 𝕍 \ D_{2}^{'}$ is a minimal inverse Nm-DS of G concerning $D_{2}^{'}$ and $D_{2}^{'} \subseteq 𝕍 \ D_{1}^{'}$ is a minimal inverse Nm-DS of G concerning $D_{1}^{'}$ . Therefore, $(N Δ)_{𝕍}^{- m} (G) > 0 .$

Conversely, consider that $(N Δ)_{𝕍}^{- m} (G) > 0,$ $D_{n}^{m}$ is a minimal Nm-DS and $D_{n}^{- m}$ is a minimal inverse Nm-DS concerning $D_{n}^{m}$ of G. Then it is clear that $D_{n}^{m}$ and $D_{n}^{- m}$ distinct. □

4 Application

PyFG has been offered as an effective tool for proper handling and modeling of the situation in conditions of uncertainty and vague information in the real world. For instance, if a decision-maker achieves a membership degree of 0.6 and a non-membership degree of 0.7 in his evaluation, this situation cannot be represented with intuitionists fuzzy graph theory because 0.6 + 0.7 > 1 . With these conditions, it is easy to see that (0.6) ² + (0.7) ² ⪯ 1 . Thus the PyFG is able to display information evaluating and modeling such a situation. In other words, Pythagorean fuzzy sets and Pythagorean fuzzy graphs have more power to administer and model the position in uncertain and vague situations. In this article, we offered the idea of a inverse Nm-DS in PyFG theory. The inverse Nm-DS in the pythagorean fuzzy network can be used to dissolve much actual issues. According to the notion of inverse Nm-DS, it can be said that inverse Nn-DSs have a supportive role in guiding and monitoring pythagorean fuzzy networks.

4.1 The role of the inverse NDS in decision making based on influencing factors the efficiency of the company

PyFG models are considered as effective, useful and widely used models in diverse fields because they show more pliability than diverse types of fuzzy graph models in delating with actual issues. Monitoring activities and making decisions at different levels of the company with favorable efficiency criteria and predetermined efficiency standards in the ashy situations betwixt certitude and incertitude, performs a significant role in increasing the efficiency and effectuality of a company. The set of influencing factors the efficiency of an company can be assumed as a PyFG.

We describe the ψ-strength and φ-strength values in any node and arc (path) as follows. For any r, s ∈ V and rs ∈ E, we possess:

$ψ_{ℙ} (r)$ : The heaviness of the direct efficacy of factor r on the efficiency of the company.

$φ_{ℙ} (r)$ : The heaviness of the indirect efficacy of factor r on the efficiency of the company.

$ψ_{ℚ} (rs)$ : The heaviness of direct influence rs on the efficiency of the company.

$φ_{ℚ} (rs)$ : The heaviness of indirect influence rs on the efficiency of the company.

In this instance, the next relationships seem logical: $\begin{matrix} ψ_{ℚ} (rs) ⪯ min {ψ_{ℙ} (r), ψ_{ℙ} (s)}, \\ φ_{ℚ} (rs) ⪯ max {φ_{ℙ} (r), φ_{ℙ} (s)} . \end{matrix}$ The relationship betwixt r and s is normal (normal arc) when the sum of the squares of the direct and indirect impacts of rs is equal to 1. Accordingly, the NDS of this graph consist factors that other factors are normally dominated with at least one of the factors of this set. Actually, the normal dominating set gives a chance for officials and chiefs of the company to concentrate on the factors of the NDS and align decisions with these factors in lieu of paying attention and monitoring a large number of decision factors. This aids company chiefs and officials make the best decisions with the uttermost sureness in the shortest possible time. For instance, Figure 3, indicates the graph of influencing factors the efficiency of an company, in which the set of D = {c, d} is a minimal NDS(with minimum node cardinality 1.45) and the set of {a, b, e, f} is an inverse NDS of G with respect to D (with node cardinality 3.04). In other words, in lieu of monitoring the six factors, only factors c, d can be monitored and paid attention and be relatively safe about advisable efficiency in the decision-making process. It is noteworthy that some factors such as common computational indices betwixt two factors, dependent calculation formula, and relevance betwixt the variables of calculating the indices of the factors perform a notable role in creating an efficacious relevance betwixt the factors. For instance, in Figure 3, demonstrates the optimal normal heaviness of the factor graph(S (F) where F is the set of all normal arcs of G) is 2.4 on advisable efficiency attainment. As we have seen in [7], it is possible to enhancement the optimal normal heaviness of the factor the graph on advisable efficiency attainment with reinforcing the relation betwixt the factors of the normal dominating set that leads to enhancement exactitude and sureness in the decision-making process and diminish the node cardinality of the NDS. Now, if it is not possible to strengthen the relationship betwixt the factors of the NDS, the inverse NDS can be used and with strengthening the relationship betwixt the factors of the NDS, the optimal normal heaviness of the factor graph can be increased. While the node cardinality of minimal NDS is also fixed, in some cases it also reduces the normal domination number of the factor graph on advisable efficiency attainment with replacing the minimal NDS. For example, in Figure 4, with establishing an normal relationship with the coordinates (0.6, 0.8) betwixt the factors a and e of $D_{n}^{- 1}$ , the normal heaviness of the graph increases to 3.0 while the minimal normal dominating set is still constant and also normal domination number of the factor graph decreased (See Table 1).

Fig. 3

PyFG G.

Fig. 4

$D_{n}^{- 1}$ .

Table 1

D	$D_{n}^{- 1}$	O (D)	$O (D_{n}^{- 1})$	$(N Δ)_{𝕍} (G)$	S (F)
D = {c, d}	$D_{n}^{- 1} = {a, b, e, f}$	1.45	3.04	2.245	2.4
D = {c, d}	$D_{n}^{- 1} = {a, b, f}$	1.45	2.39	1.92	3.0
D = {c, d}	$D_{n}^{- 1} = {b, e, f}$	1.45	2.19	1.82	3.0

5 Conclusion

Many issues of practical interest can be represented with the graphs. The notion of domination in graph is notable in theoretical developments and usage. A model based on the Pythagorean fuzzy set has more pliability to deal with human evaluation information than other fuzzy and vague models. In other words, Pythagorean fuzzy sets and PyFGs have more power to administer and model the situation in uncertain and vague situations. In this article, we defined for the first time the notions of Nm-DS and inverse Nm-DS in a PyFG. Finally, by using the notion of inverse NDS and the diminution efficacy of an additional normal arc on the normal domination number parameter, a model for optimizing the normal domination parameter was provided. The advantage of this model over the model presented in [7] is that optimized the normal domination number parameter in specific and targeted sections more accurately while ensuring the stability or reduction of the normal domination number parameter in the Pythagorean fuzzy network.

Footnotes

Acknowledgment

The authors are grateful to the referees for their valuable comments and suggestions, which have greatly improved the paper.

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