Pythagorean Dombi fuzzy graphs

Pythagorean fuzzy graph, a broadly used extension of fuzzy and intuitionistic fuzzy graph, is helpful in representing structural relationships between several objects where the relation between these objects is vague, while the Dombi operators with operational parameters have excellent flexibility. Utilizing these two concepts, this research paper proposes the novel concept of Pythagorean Dombi fuzzy graphs (PDFGs). Basically, graph terminology is employed for introducing Pythagorean fuzzy analogs of various fundamental graphical ideas using Dombi operator. Further, under Pythagorean Dombi fuzzy environment, regular, totally regular, strongly regular and biregular graphs are defined with appropriate illustration and some of their crucial properties are examined. Meanwhile, the notion of edge regularity of PDFG is also initiated with substantial characteristics. Finally, a numerical example related to evaluation of appropriate ETL software for a business intelligence project is presented to better understand PDFGs.


Introduction
In the past few years, many operators were initiated and the most important among them appeared in numerous monographs concerning with fuzzy logic.Particularly, it includes min-max, Einstein, Hamacher, Frank, product, Lukasiewicz, Azcel-Alsina and Dombi operators.The product and Einstein operators are the special cases of Hamacher operator.From practical perspectives, these parametrical families hold one's attention because by considering different value of parameter a distinct argument can be formed.
Zadeh [1] recommended to utilize the product and the minimum operator for defining fuzzy set.Hamacher [2] emphasised that by taking the solution of associativity operational equality, these operators can be created.Later, he obtained the rational structure of disjunctive and conjunctive operators in accordance with Kuwagaki's results [3].From that moment on, the researchers working in the field of fuzzy theory proposed a more general form, i.e., triangular norms (t-norms) and triangular conorms (t-conorms).Menger [4] introduced t-norms and t-conorms within probabilistic metric framework, where numbers are employed to narrate the distance between two objects of the space.Schweizer and Sklar [5] presented many axioms and results related to t-norms and t-conorms that showed the rapid progress of this field.Furthermore, these norms are certified as standard models for union and intersection of fuzzy sets by Alsina et al. [6].Several extensions and summarizations of beneficial outcomes of T -operators for the similar cause can be seen in Klement et al. [7] and [8], respectively.In every fuzzy logic application, specifically fuzzy graph theory and decision-making procedures, Zadeh's min and max operators have been extensively applied.From experimental and theoretical point of view, other T -operators may perform better in some cases, especially in decision-making problems, such as product operator may tend to choose over min operator [9].One has to observe and examine the characteristics of T -operators like suitability of the model, simplicity and implementation of hardware and software before the appropriate selection of these operators for a stated application.Since the study and work on these operators has broadened, multiple choices are available for choosing T -operators that may be preferable for given research.
A pictorial representation that bonds the items together is known as 'graph'.But if in the bonding there occurs haziness, then the graph can be considered as fuzzy graph.Rosenfeld [10] established the layout of fuzzy graphs by taking into account fuzzy relations on fuzzy sets (FSs) with min and max operators.As the hesitancy part was not characterized explicitly, Atanassov [11] extended fuzzy sets to intuitionistic fuzzy sets (IFSs) by allocating membership μ and non-membership grade ν to the items, satisfying the constrain μ + ν ≤ 1 with hesitancy part π = 1 − μ − ν.At that time graph theory was widely applied in almost every field of real life; hence Shannon and Atanassov [12] put forward the idea of intuitionistic fuzzy graphs by considering intuitionistic fuzzy relations on IFSs.Yager [13][14][15] inaugurated Pythagorean fuzzy sets (PFSs), a new extension of IFSs, to manage the complex uncertainty and impreciseness with constrain μ 2 + ν 2 ≤ 1, where μ and ν represent membership and non-membership grade, respectively.Afterward, for explicating the dual features of an item, Zhang and Xu [16] proposed the notion of Pythagorean fuzzy number (PFN).The motivation of PFSs can be described as follows: in a decision-making environment, a specialist gives the preference information about an alternative with the membership grade 0.6 and the non-membership grade 0.5; it is noted that the intuitionistic fuzzy number (IFN) fails to address this situation, as 0.6 + 0.5 > 1.But (0.6) 2 + (0.5) 2 ≤ 1. Hence PFSs comprise and accommodate greater amount of vagueness than IFSs.The comparison between IFN space and PFN space is shown in Fig. 1.It has been successfully applied in numerous areas, including the internet stocks investment [16], the service quality of domestic airline [17] and the governor selection of the Asian Investment Bank [18].In the practical multi-criteria group decision-making problems, Akram et al. [19] showed that PFS has much stronger ability to model fuzziness.Under PF environment, many researchers have initiated work in different directions and acquired various eminent results [17].Some operations on PFSs [20] and Pythagorean fuzzy TODIM approach to multi-criteria decision making [18] have been discussed.Furthermore, the PFS has been investigated from different perspectives, including aggregation operators [21,22].Garg [23][24][25][26] explored applications of Pythagorean fuzzy sets in decision-making problems.Graph theory has several applications, including cluster analysis and optimization of networks.Hence on the basis of its applications, Naz et al. [27] proffered the idea of Pythagorean fuzzy graphs (PFGs) using min and max operators and holding Pythagorean fuzzy relations on PFSs.Verma et al. [28] opened up the concept of strong Pythagorean fuzzy graphs and defined complement as well.Energy under Pythagorean fuzzy environment was Fig. 1 Comparison of spaces of the IFN and the PFN discussed by Akram and Naz [29].Akram et al. [19,[30][31][32] proposed certain graphs and explored their crucial properties under Pythagorean fuzzy circumstances.Recently, Akram and Habib [33] discussed regularity of q-rung picture fuzzy graphs with applications.Habib et al. [34] presented the notion of q-Rung orthopair fuzzy competition graphs by considering the most wide spread max and min operators and gave an application in the soil ecosystem.Dombi [35] inaugurated Dombi operator with flexible operational parameter in 1982.This operator is exceptional as the sign of the parameter discovers whether the operator type is disjunctive or conjunctive.In decision-making problems, it is very useful as by taking distinct values of operational parameters, different arguments can be made depending upon the requirement or one's need.For this precedence, Chen and Ye [36], Jana et al. [37], Shi and Ye [38] used Dombi operations and presented MCDM problem in singlevalued neutrosophic, neutrosophic cubic and bipolar fuzzy environment, respectively.Liu et al. [39] proffered MCGDM problem utilizing Dombi Bonferroni mean operator on IFSs.He [40] investigated typhoon disaster assessment by considering Dombi operations in hesitant fuzzy environment.From the existing studies, it is observed that Dombi operational parameters have flexible nature in decision-making areas.Fuzzy graph theory can easily structure and model decision-making problems with uncertainty.A very limited effort is made for using Dombi operator in the field of graph theory.Hence on the basis of it, Ashraf et al. [41] presented the idea of Dombi fuzzy graph (DFG).As Pythagorean fuzzy graph is more powerful and more practical tool having the capability to deal with imprecise and incomplete information in different decision-making disciplines, such as engineer-ing, mathematics, statistics, artificial intelligence, medical and social sciences than fuzzy graphs.Therefore, the main objective of this research article was to emphasize on the fact that for generalizing the classical graphs to Pythagorean fuzzy graphs, the min and max operators are not preferable to model certain real-world situations.Further, our aim was to assemble the development introduced by Klement, Alsina, Hamachar and other founders in the field of Pythagorean fuzzy graph theory.The paper is accumulated to demonstrate the usage of a T -operator, especially the Dombi operator.As Dombi operators with operational parameters, have excellent flexibility and have not yet been applied for Pythagorean fuzzy graphs, hence motivated from these operators, this paper introduces the notion of PDFGs as a generalization of Dombi fuzzy graphs.Further, some substantial characteristics like strongness, completeness, vertex and edge regularity are inspected as they are extensively applied in designing reliable computer networks and matrix representations.
The presented research article is structured as follows: Sect. 2 describes the basic notions and terminologies which will be utilized in the rest of sections.In Sect.3, we propose the novel concept of Pythagorean Dombi fuzzy graphs and define the complement, homomorphism, isomorphism, completeness and strongness with appropriate illustration.In Sect.4, we introduce regular, totally regular, strongly regular, biregular, edge regular, totally edge regular Pythagorean Dombi fuzzy graphs and examine some of their crucial characteristics.Section 5 presents a decision-making algorithm in Pythagorean Dombi fuzzy environment and solves a numerical example to illustrate the developed method.Section 6 contains concluding remarks and points out directions for future work.

Preliminaries
In this section, some prerequisite notations and terminologies have been stated for better understating.
In crisp sense, a graph is an ordered pair G * = (V , E), where V is a vertex set and E is the edge set of G * .A vertex connected by an edge to a vertex y is called a neighbor of y.The number of edges incident to a vertex y is called the degree of that vertex.A graph without loops and multiple edges is called simple graph, whereas a graph in which each pair of graph vertices is connected by an edge is known as complete graph.The complement G * of a graph, G * , is a graph having vertex set same as in G * , in which two vertices are incident if and only if they are not incident in G * .If there occurs a one-one correspondence between the vertices of two graphs )) which preserves adjacency, then the graphs G * 1 and G * 2 are called isomorphic.A self-complementary graph is a graph which is isomorphic to its complement.A regular and biregular graph is a graph, where each vertex has the same number of neighbors and each two vertices on the same side of the given bipartition have the same degree as each other, respectively.A graph with direction is called directed graph, where without direction is known as undirected graph.
Definition 1 [1] A fuzzy set (FS) on a universe X is an object of the following form: Definition 2 [10] A fuzzy set on X ×X is said to be a fuzzy relation (FR) on X , denoted by Definition 3 [10] A fuzzy graph on a non-empty set X is a pair G = (A , B) with A a FS on X and B a FR on X such that for all s, t ∈ X , where A : X −→ [0, 1] and B : Definition 4 [11] An intuitionistic fuzzy set (IFS) on a universe X is an object of the form where μ A : X −→ [0, 1] and ν A : X −→ [0, 1] represent the membership and non-membership grades of A , and μ A , ν A satisfies the condition 0 ≤ μ A (s)+ν A (s) ≤ 1 for all s ∈ X .
Definition 6 [12] An intuitionistic fuzzy graph (IFG) on a non-empty set X is a pair G = (A , B) with A an IFS on X and B an IFR on X such that and 0 ≤ μ B (st) + ν B (st) ≤ 1 for all s, t ∈ X , where μ B : X × X −→ [0, 1] and ν B : X × X −→ [0, 1] represent the membership and non-membership grades of B, respectively.
Definition 7 [13] A Pythagorean fuzzy set (PFS) on a universe X is an object of the form where μ A : X −→ [0, 1] and ν A : X −→ [0, 1] represent the membership and non-membership grades of A , and Definition 8 [27] A Pythagorean fuzzy set on X ×X is said to be a Pythagorean fuzzy relation (PFR) on X , denoted by represent the membership and non-membership grades of B, respectively, such that 0 ≤ μ 2 B (st) + ν 2 B (st) ≤ 1 for all s, t ∈ X .
Definition 9 [27] A Pythagorean fuzzy graph (PFG) on a non-empty set X is a pair G = (A , B) with A a PFS on X and B a PFR on X such that represent the membership and non-membership grades of B, respectively.
Replacing 1 by 0 in condition 1, we obtain the concept of dual t-conorm or s-norm.
There are the following common t-norms: -The minimum operator M (s, t) = min(s, t).
-The product operator P(s, t) = st.
-The Dombi's t-conorm One Definition 11 [41] A Dombi fuzzy graph on underlying set V is an ordered pair G = (A , B), where A : for all s, t ∈ V , where μ A and μ B represent the membership grades of A and B, respectively.
Definition 12 [27] A Pythagorean fuzzy preference relation (PFPR) on the set of alternatives X = {x 1 , x 2 , . . ., x n } is represented by a matrix Q = (r i j ) n×n , where r i j = (x i x j , μ(x i x j ), ν(x i x j )) for all i, j = 1, 2, . . ., n.For convenience, let r i j = (μ i j , ν i j ) where μ i j indicates the degree to which the object x i is preferred to the object x j , ν i j denotes the degree to which the object x i is not preferred to the object x j and π i j = 1 − μ 2 i j − ν 2 i j is interpreted as a hesitancy degree, with the following conditions: For other applications and prerequisite terminologies, the readers are referred to [42][43][44][45][46][47][48].

Pythagorean Dombi fuzzy graphs
Definition 13 A Pythagorean Dombi fuzzy graph (PDFG) on underlying set V is an ordered pair G = (A , B), where and 0 ≤ μ 2 B (st) + ν 2 B (st) ≤ 1 for all s, t ∈ V , where μ B and ν B represent the membership and non-membership grades of B, respectively.

Remark 1
We call A the Pythagorean Dombi fuzzy vertex set of G and B the Pythagorean Dombi fuzzy edge set of G.
and defined by (D) G (s) = ((D) μ (s), (D) ν (s)), where -The total degree of vertex s ∈ V is symbolized by , where Example 3 From Example 1, we have -The degree of the vertices in G are as follows: -The total degree of the vertices in G are as follows: Definition 16 The complement of a PDFG G = (A , B) on an underlying graph By routine computations, one can see from Fig.
Proof Suppose that G is a PDFG.Then by definition of complement of PDFG, we have for all s, t ∈ V .Hence the complement of a complement PDFG is a PDFG itself.
Definition 18 An isomorphism H : Proof Assume that G is a self-complementary PDFG; then there occurs an isomorphism Using the definition of complement of G, we have Likewise, for non-membership grade, we have This completes the proof.
Proof Assume that G is the PDFG that satisfies for all s, t ∈ V , then the identity mapping I : V −→ V is an isomorphism from G to G that satisfies the following conditions: Since the membership grade of an edge st is given by for all s, t ∈ V , we have Likewise, the non-membership grade of an edge st is given by for all s, t ∈ V ; so we have ) are two isomorphic PDFGs, then the complement of G 1 and G 2 are also isomorphic to each other and conversely.
Proof Assume that G 1 and G 2 are two isomorphic PDFGs.Then by definition of isomorphism, there occurs a bijective mapping By using the definition of complement of PDFG, the membership grade of an edge st is Also, the non-membership grade of an edge st is Hence the complement of G 1 is isomorphic to the complement of G 2 .Likewise, the converse part can be proved.
) are two weak isomorphic PDFGs, then the complement of G 1 and G 2 are also weak isomorphic.
Proof Assume that G 1 and G 2 are two weak isomorphic PDFGs.Then by definition of weak isomorphism, there occurs a bijective mapping Consider the membership condition of an edge; then we have Now take the non-membership condition of an edge; then we have Hence we conclude that G 1 and G 2 are weak isomorphic. 123

Fig. 5 Complete Pythagorean Dombi fuzzy graph
We state the following proposition without proof: By routine computations, one can see from Fig. 5 that G = (A , B) is a complete PDFG.
Definition 23 A PDFG is said to be strong if ,     6. Then by using the Definition 24, one can obtain complement of a strong PDFG as presented in Fig. 7.
By routine computations, one can see from Fig. 7 that G is a PDFG.

Regularity of Pythagorean Dombi fuzzy graphs
In this section, we introduce the concept of regularity of Pythagorean Dombi fuzzy graphs that can be helpful in real scientific applications.8, is (0.82, 0.92)-regular PDFG.
Proof Assume that G is a PDFG with size given by Since G is (R 1 , R 2 )-regular, the degree of the vertex Since each edge is double times counted, one time for vertex s and one time for t, we have This completes the proof.
Proof Assume that G is a PDFG; then the total degree is given by Since each edge is double times counted, one time for vertex s and one time for t, we have This completes the proof.
As G is a (K 1 , K 2 )-totally regular PDFG, from Theorem 3, we must have This completes the proof.
-regular PDFG; then the degree of each vertex in G 1 is given by

Theorem 5 Let G = (A , B) be a PDFG on underlying graph G * = (V , E). If μ A and ν A are constant functions, then following statements are equal:
1. G = (A , B) is a regular PDFG.

Proof Assume that μ A and ν
So the total degree of vertex is given by Hence G is a regular PDFG.Therefore, (1) ⇒ ( 2) is proved.Now suppose that G is a (K 1 , K 2 )-totally regular PDFG, then Likewise, for non-membership grade So G is a totally regular PDFG.Therefore, (2) ⇒ (1) is proved.Hence we conclude that (1) and (2) are equal.
-regular as well as (K 1 , K 2 )-totally regular PDFG, then μ A and ν A are constant functions.
Proof Assume that G is a (R 1 , R 2 )-regular and (K 1 , K 2 )totally regular PDFG.Then the degree of vertex is and the total degree of vertex is Further, it follows that Likewise, for non-membership grade

Hence we conclude that μ
Further, the adjoining and non-adjoining vertices have equal common neighbourhood As G is a complete PDFG, we must have Hence G is a ((n − 1)C 3 , (n − 1)C 4 )-regular PDFG.Further, the sum of the membership and non-membership grades of the common neighbouring vertices of any pair of adjoin- As G is complete, the sum of the membership and nonmembership grades of the common neighbouring vertices of any pair of non-adjoining vertices L = (L 1 , L 2 ) = (0, 0) are equal.Since all the properties are satisfied, we conclude that G = (A , B) is a strongly regular PDFG.
) is a strongly regular disconnected PDFG, then the sum of the membership and non-membership grades of the common neighbouring vertices of any pair of non-adjoining vertices is L = (L 1 , L 2 ) = (0, 0).
) is a strongly regular complete bipartite PDFG with equal bipartition, then the sum of the membership and non-membership grades of the common neighbouring vertices of any pair of non-adjoining vertices is L = (L 1 , L 2 ) = (0, 0).
Proof Assume that G is strongly regular PDFG; then by definition G is (R 1 , R 2 )-regular.Further, as G is strong, then G is also strong.Therefore, we must have In G, the degree of a vertex s j is defined by

Theorem 9 Consider G = (A , B) be a strong PDFG; then G is a strongly regular PDFG if and only if G is a strongly regular PDFG.
Proof Suppose that G is a strongly regular PDFG; then by definition, G is (R 1 , R 2 )-regular.Further, the adjoining and non-adjoining vertices have equal common neighbourhood U = (U 1 , U 2 ) and L = (L 1 , L 2 ), respectively.To prove that G is a strongly regular PDFG, we must show that G is (R 1 , R 2 )-regular.Since G is a strongly regular and strong, then by Theorem 8, G is (R 1 , R 2 )-regular.Moreover, assume that S 1 = {(s j , s k )/(s j , s k ) ∈ E} and S 2 = {(s j , s k )/(s j , s k ) / ∈ E} be the sets of all adjoining and non-adjoining vertices of G, where s j and s k have equal ∈ E}, where s j and s k have equal common neighbourhood L = (L 1 , L 2 ) and U = (U 1 , U 2 ), respectively.Hence G is strongly regular PDFG.
Conversely, G is a strongly regular PDFG; then by definition, G is (R 1 , R 2 )regular.Further, the adjoining and non-adjoining vertices have equal common neighbourhood L = (L 1 , L 2 ) and U = (U 1 , U 2 ), respectively.To prove that G is a strongly regular PDFG, we must show that G is (R 1 , R 2 )-regular.Since G is a strongly regular and strong, then by Corollary 2, G is (R 1 , R 2 )-regular.Furthermore, assume that S 1 = {(s j , s k )/(s j , s k ) ∈ E} and S 2 = {(s j , s k )/(s j , s k ) / ∈ E} be the sets of all adjoining and non-adjoining vertices of G, where s j and s k have equal common neighbourhood: ∈ E}, where s j and s k have equal common neighbourhood U = (U 1 , U 2 ) and L = (L 1 , L 2 ), respectively.Hence G is strongly regular PDFG.
Then the degree of the vertex s is given by Likewise, since μ A and ν A are constant functions, the parameters are (UC 1 , UC 2 ) and (LC 1 , LC 2 ).Hence G is a strongly regular PDFG with parameters (RC 3 , RC 4 ), (UC 1 , UC 2 ) and (LC 1 , LC 2 ).
Remark 6 Converse of the Theorem 10 need not be true as seen in the example given below: Consider a graph G * = (V , E), where V = {t 1 , t 2 , t 3 , t 4 } and E = {t 1 t 2 , t 1 t 3 , t 1 t 4 , t 2 t 3 , t 2 t 4 , t 3 t 4 }.Let A and B be Pythagorean Dombi fuzzy vertex set and Pythagorean Dombi fuzzy edge set defined on V and E, respectively.By routine computations, one can see from Fig. 13 that G is a strongly regular PDFG with parameters R = (0.73, 2.19), U = (1, 0.12) and L = (0, 0).Meanwhile, G * is a strongly regular graph with parameter (4, 3, 2, 0) and μ A , ν A are constant functions.But μ B , ν B are not constant functions.Theorem 11 Let G be a (n, R, U, L) strongly regular PDFG; then the degree sequence of n elements of G is a constant sequence Proof Assume that G is a (n, R, U, L) strongly regular PDFG; then by definition, G is a (R 1 , R 2 )-regular.Thus all the vertices have same degree Theorem 12 Let G be a (n, R, U, L) strongly regular PDFG; then the degree set of the membership and non-membership grades of G is a singleton set {R 1 } and {R 2 }, respectively.Proof Assume that G is a (n, R, U, L) strongly regular PDFG; then by definition, G is a (R 1 , R 2 )-regular.Thus all the vertices have same degree (D) G (s i ) = (R 1 , R 2 ), where i = 1, 2, . . ., n.As the degree sequence of G is a constant sequence (R 1 , R 1 , . . ., R 1 ; R 2 , R 2 , . . ., R 2 ), then the corresponding membership and non-membership degree set is {R 1 } and {R 2 }, respectively.By routine computations, see from Fig. 15 that G has constant membership and non-membership degree sequence (0.5, 0.5, 0.5) and (1.3, 1.3, 1.3), respectively, whereas, the corresponding membership and nonmembership degree set is {0.5} and {1.3}, respectively.But G is not strongly regular PDFG as the sum of the membership and non-membership grades of the common neighbouring vertices of any pair of adjoining vertices of G are not equal.
• The total degree of an edge st ∈ E is symbolised by (T D) G (st) and defined by (T D) G (st) = ((T D) μ (st), (T D) ν (st)), where Example 13 Consider a PDFG G over V = {t Proof Assume that G is a (R 1 , R 2 )-edge regular and (K 1 , K 2 )-totally edge regular PDFG.Then the degree of edge is and the total degree of edge is Further, it follows that Likewise, for non-membership grade Hence we conclude that μ B = K Proof Assume that G is a PDFG on a regular crisp graph G * .Also, suppose that μ B and ν B are constant functions; then μ B (s j s k ) = C 1 and ν B (s j s k ) = C 2 for all s j s k ∈ E. By using the definition of vertex degree (D) G (s j ) = ((D) μ (s j ), (D) ν (s j )), we must have Further, by definition of total degree of edge, we have In the similar manner, we can easily show that (T D) ν (s j s k ) = K 2 for all s j s k ∈ E. Hence we conclude that G = (A , B) is both (R 1 , R 2 )-regular and (K 1 , K 2 )-totally edge regular PDFG.
Conversely, assume that G is both (R 1 , R 2 )-regular and (K 1 , K 2 )-totally edge regular PDFG.To prove that μ B and ν B are constant functions, consider the definition of total degree of edge, we have In the similar manner, we can easily show that μ B (s j s k ) = 2R 2 − K 2 for all s j s k ∈ E. Hence μ B and ν B are constant functions.

Theorem 17 Let G = (A , B) be a PDFG on a crisp graph G
Proof Suppose that μ B and ν B are constant functions such that μ B (s j s k ) = C 1 and ν B (s j s k ) = C 2 for all s j s k ∈ E. Assume that G is (R 1 , R 2 )-edge regular.To prove that G * is an edge regular graph, we suppose on contrary that G * is not an edge regular graph, i.e., (D) G * (s j s k ) = (D) G * (s l s n ) for at least one pair of s j s k , s l s n ∈ E. By definition of degree of edge of PDFG, we have

Likewise, we can easily show that
edge regular PDFG, a contradiction.Hence we conclude that G * is an edge regular graph.
Conversely, assume that G * is an edge regular graph.To show that G is (R 1 , R 2 )-edge regular PDFG, we suppose on the contrary that G is not (R 1 , R 2 )-edge regular PDFG, i.e., (D) G (s j s k ) = (D) G (s l s n ) for at least one pair of s j s k , s l s n ∈ E, ((D) μ (s j s k ), (D) ν (s j s k )) = ((D) μ (s l s n ), (D) ν (s l s n )).Now (D) μ (s j s k ) = (D) μ (s l s n ) implies that

Numerical example to decision-making
In this section, a decision-making problem concerning the evaluation of appropriate ETL (Extract, Transform and Load) software for a Business Intelligence (BI) project (adopted from [47]) is solved to illustrate the applicability of the proposed concept of PDFGs in realistic scenario.The algorithm for the evaluation of appropriate ETL software for a BI project within the framework of Pythagorean fuzzy preference relation (PFPR) [27] is outlined in Algorithm 1.

Evaluation of appropriate ETL software for a BI project
Business Intelligence, a field of information systems architecture, allows implementation that includes collection, transformation and restoring of data for assisting the decisionmaking experts in enterprises.The central part of BI is established on data warehouses powered by ETL.With the gradual development of BI usage, ETL, the initial point of the project, has become a key factor that affects the failure or success of the BI project.The main task of BI project is the evaluation of most appropriate and suitable ETL software which maximizes the profits, limits the costs, performs well and is flexible to accommodate future advancements in the project.A number of ETL software are available in the market.Each software has its own technique for extracting, loading and transforming of data.A decision-making expert is hired that pairwise compares the five ETL softwares S l (l = 1, 2, . . ., 5) for a new BI project on the basis of the criterion 'functionality and reliability' and provides his preference information in the form of PFPR Q = (q lp ) 5×5 , where q lp = (μ lp , ν lp ) is the Pythagorean fuzzy element assigned by the decision-making expert with μ lp as the degree to which the ETL software S l is preferred over the ETL software S p with respect to the given criterion and ν lp as the degree to which the ETL software S l is not preferred over the ETL software S p with respect to the given criterion.The PFPR Q = (q lp ) 5×5 is expressed in the following tabular form (Table 1).The Pythagorean fuzzy directed network D corresponding to PFPR Q given in Table 1, is presented in Fig. 20.
On the basis of score functions, we get the ranking of the ETL softwares S l , l = 1, 2, . . ., 5 as follows: According to the ranking, it is concluded that S 3 is the most appropriate ETL software for a new project among all.
On the basis of score functions, we get the ranking of the ETL softwares S l , l = 1, 2, . . ., 5 as follows: According to the ranking, it is concluded that S 3 is the most appropriate ETL software for a new project among all We present our proposed method for decision-making in the following Algorithm 1:

Definition 24
The complement of a strong PDFG G = (A , B) on an underlying graph G

Fig. 7 Example 7
Fig. 7 Complement of a strong Pythagorean Dombi fuzzy graph

1 −
R 1 and ν B = K 2 − R 2 are constant functions.Theorem 16 Let G = (A , B) be a PDFG on a regular crisp graph G * = (V , E).Then μ B and ν B are constant functions if and only if G = (A , B) is both (R 1 , R 2 )-regular and (K 1 , K 2 )-totally edge regular PDFG.
s j s p ∈E, p =k μ B (s j s p ) + s k s p ∈E, p = j μ B (s k s p ) = s l s m ∈E,m =n μ B (s l s m ) + s n s m ∈E,m =l μ B (s n s m ).As μ B is a constant function, we have (D) G * (s j s k ) = (D) G * (s l s n ), a contradiction.Hence we conclude that G is a (R 1 , R 2 )-edge regular PDFG.
By routine computations, one can see from Fig.6 that G = (A , B) is a strong PDFG.Every complete PDFG is strong; one can easily see from Example 5.
t 3 , t 4 , t 5 } and E = {t 1 t 2 , t 1 t 4 , t 1 t 5 , t 2 t 3 , t 2 t 4 , t 3 t 4 , t 4 t 5 }.Let A and B be Pythagorean Dombi fuzzy vertex set and Pythagorean Dombi fuzzy edge set defined on V and E, respectively.
t 3 , t 4 } and E = {t 1 t 3 , t 1 t 4 , t 2 t 3 , t 2 t 4 }.Let A and B be Pythagorean Dombi fuzzy vertex set and Pythagorean Dombi fuzzy edge set defined on V and E, respectively.
R 2 are constant functions.= {t 1 t 2 , t 2 t 3 }.Let A and B be Pythagorean Dombi fuzzy vertex set and Pythagorean Dombi fuzzy edge set defined on V and E, respectively.
* = (V , E), where V = {t 1 , t 2 , t 3 } and E Since μ A (t i the sum of the membership and non-membership grades of the common neighbouring vertices of any pair of adjoining vertices of G are equal and represented as U = (U 1 , U 2 ); 3. the sum of the membership and non-membership grades of the common neighbouring vertices of any pair of nonadjoining vertices of G are equal and represented as L = (L 1 , L 2 ).

Remark 7
Converse of the Theorems 11 and 12 need not be true as seen in the example given below.Consider a graph G * = (V , E), where V = {t 1 , t 2 , t 3 } and E = {t 1 t 2 , t 1 t 3 , t 2 t 3 }.Let A and B be Pythagorean Dombi fuzzy vertex set and Pythagorean Dombi fuzzy edge set defined on V and E, respectively.

Table 1
PFPR of the decision-making expert