Plane Thermoelastic Waves in Infinite Half-Space Caused Decision Making: Applications in Management and Engineering Vol. 3, Issue 2, 2020, pp. 119-130 ISSN: 2560-6018 eISSN: 2620-0104 DOI: https://doi.org/10.31181/dmame2003119m * Corresponding author. E-mail addresses: km.18ma1103@phd.nitdgp.ac.in (K. Mohanta), arindam84nit@gmail.com (A. Dey), anita.buie@gmail.com (A. Pal) A STUDY ON PICTURE DOMBI FUZZY GRAPH Kartick Mohanta 1*, Arindam Dey 2 and Anita Pal 1 1 Department of Mathematics, National Institute of Technology Durgapur, India 2 Department of Computer Sciences and Engineering, Saroj Mohan Institute of Technology, Hooghly, India Received: 3 June 2020; Accepted: 4 September 2020; Available online: 20 September 2020. Original scientific paper Abstract: The picture fuzzy graph is a newly introduced fuzzy graph model to handle with uncertain real scenarios, in which simple fuzzy graph and intuitionistic fuzzy graph may fail to model those problems properly. The picture fuzzy graph is used efficiently in real world scenarios which involve several answers to these types: yes, no, abstain and refusal. In this paper, the new idea of dombi picture fuzzy graph is introduced. We also describe some operations on dombi picture graphs, viz. union, join, composition and cartesian product. In addition, we investigated many interesting results regarding the operations. The concept of complement and isomorphism of Picture dombi fuzzy graph are presented in this paper. Some important results on weak and co-weak isomorphism of Picture dombi fuzzy graph are derived. Key words: t-norm, s-norm, Picture Dombi Fuzzy Graph, Union, Composition, Cartesian Product, Join, Complement, Homomorphism, Isomorphism. 1. Introduction Menger (1942) presented triangular norms (t-norms) and triangular co-norms (t- conorms) in the framework of probabilistic metric spaces which were later defined and discussed by Schweizer and Berthold (2011). Alsina et al. (1983) proved that t-norms and t-conorms are standard models for intersecting and unifying fuzzy sets, respectively. Since then, many other researchers have presented various types of T-operators for the same purpose (Hamacher, 1978). Zadeh’s conventional T-operators, min and max, have been used in almost every application of fuzzy logic particularly in decision-making processes and fuzzy graph theory. It is a well-known fact that from theoretical and experimental aspects other T-operators may work better in some situations, especially in the context of decision-making processes. For example, the product operator may be preferred to the min operator (Dubois et al., 2000). For the selection of appropriate T- mailto:km.18ma1103@phd.nitdgp.ac.in mailto:arindam84nit@gmail.com mailto:anita.buie@gmail.com Mohanta et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 119-130 120 operators for a given application, one has to consider the properties they possess, their suitability to the model, their simplicity, their software and hardware implementation, etc. As the study on these operators has widened, multiple options are available for selecting T-operators that may be better suited for given research. There are various real- life problems that we cannot explain with the concept of fuzzy set theory. For solving these kinds of problem, K. Atanassov (1986) proposed the idea of an intuitionistic fuzzy set (IFS). In IFS we consider membership function and non-membership function such that their sum is lying in [0; 1]. In IFS theory, the idea of neutrality membership value is not considering. In many real-life situations, the neutral membership degree is needed, like a democratic election station. Human beings generally give opinions having more answers of the type: yes, no, abstain and refusal. For example, in a democratic voting system, 1000 people participated in the election. The election commission issues 1000 ballot paper and one person can take only one ballot for giving his/her vote and A is only one candidate. The results of the election are generally divided into four groups came with the number of ballot papers namely “vote for the candidate (500)”, “abstain in the vote (200)”, “vote against a candidate (200)” and “refusal of voting (100)”. The “abstain in the vote” describes that ballot paper is white which contradicts both “vote for the candidate” and “vote against a candidate” but it considers the vote. However, “refusal of voting” means bypassing the vote. This type of real-life scenarios cannot be handled by intuitionistic fuzzy set. If we use intuitionistic fuzzy sets to describe the above voting system, the information of voting for non-candidates may be ignored. To solve this problem, Cuong and Kreinovich (2014) proposed the concept of picture fuzzy set which is a modified version of the fuzzy set and Intuitionistic fuzzy set. Picture fuzzy set (PFS) allows the idea degree of positive membership, degree of neutral membership and degree of negative membership of an element. Graph theory is an important mathematical tool for handling many real-world problems. Graph theory has various application in different areas like computer science, social sciences, economics, physics, system analysis, chemistry, neural networks, electrical engineering, control theory, transportation, architecture, and communication. Kaufmann (1975) introduces the basic concept of fuzzy graph theory and after that Rosenfeld (1975) describes more idea on the fuzzy graph-theoretic concept. Krassimir T Atanassov introduces the concept of intuitionistic graph theory. Shovan Dogra (2015) describes different types of product of fuzzy graphs. Havare, Özge Çolakoğlu, n.d. discussed on the coronary product of two fuzzy graphs. In this paper we present the concept of Picture dombi fuzzy graph (PDFG) and discussed the operations like union, join, composition, cartesian product, h-morphism, isomorphism, complement of DPFG’s. We also introduce some theorems and examples on PDFG’s. 2. Preliminaries t-norm A t-norm is a binary mapping : [0,1] [0,1] [0,1]t   which is satisfies the following conditions: , , , [0,1]a b c d  1. (Boundedness property) (0, 0) 0, ( ,1) (1, )t t a t a a   ; 2. (Monotonicity property) ( , ) ( , )t a b t c d , if a c and b d ; 3. (Commutativity property) ( , ) ( , )t a b t b a ; 4. (Associativity property) ( , ( , )) ( ( , ), )t a t b c t t a b c . A Study on Picture Dombi Fuzzy Graph 121 t-conorm or s-norm A t-conorm is a binary mapping : [0,1] [0,1] [0,1]s   which is satisfies the following conditions: , , , [0,1]a b c d  (Boundedness property) (1,1) 1s  , ( , 0) (0, )s a s a a  ; (Monotonicity property) ( , ) ( , )s a b t c d , if a c and b d ; (Commutativity property) ( , ) ( , )s a b s b a ; (Associativity property) ( , ( , )) ( ( , ), )s a s b c s s a b c . Hamacher norm Hamacher define t-norm and s-norm as follows: , [0,1]a b  (t-norm) ( , ) (1 )( ) ab t a b a b ab       , 0  . (s-norm) ( 1) ( , ) 1 ab a b s a b ab        , 1   . Dombi norm The Dombi norm is given by , [0,1]a b  (t-norm) 1 1 ( , ) 1 1 1 [( ) ( ) ] t a b a b a b         ; (s-norm) 1 1 ( , ) 1 1 1 [( ) ( ) ] s a b a b a b            . Remark 1: If we put 1  in Dombi t-norm, we have ( , ) ab t a b a b ab    , , [0,1]a b  . If we put 1  in Dombi s-norm, we have 2 ( , ) 1 a b ab s a b ab     , , [0,1]a b  . Fuzzy set Let X be a universal set. A fuzzy set M of X is the collection of elements  in X s. t., ( ) [0,1]T   . Here T is called a membership function of M i.e., : [0,1]T X  . Fuzzy graph A f-graph of the graph ( , ) G G G V E  is a pair ( , )G    , where : [0,1]V  is a fuzzy set on G V and : [0,1] G G V V   is a fuzzy relation on G V s. t., ( , ) ( ) ( )x y x y     , ( , ) G G x y V V   (Zadeh, 1965). Picture Fuzzy set (PFS) Let be an universal set. A PFS is defined as follows { , ( ), ( ), ( ) : 0 ( ) ( ) ( ) 1, }                     . Here : [0,1]  , : [0,1]  and : [0,1]  are called positive membership degree, neutral membership degree and negative membership degree respectively. For all   , 1 ( ( ) ( ) ( ))          is called refusal function of  in . Mohanta et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 119-130 122 Picture Fuzzy Relation (PFR) Let and be two universal sets. A PFR is subset of  s. t., { ( , ), ( , ), ( , ), ( , ) : 0 ( , ) ( , ) ( , ) 1, ( , ) }                               , where : [0,1]   , : [0,1]   and : [0,1]   are called positive membership function, neutral membership function and negative membership function respectively. Dombi Graph Let ( , ) G G G V E be a crisp undirected graph contain no self-loop and parallel edges. Also, let : [0,1]V  membership degree on V and : [0,1]V V   be the membership degree on the symmetric fuzzy relation E V V  . Then ( , , )V   , is said to be a dombi graph if ( ) ( ) ( , ) ( ) ( ) ( ) ( ) a b a b a b a b           , ( )ab E  . Picture Dombi Fuzzy Graph (PDFG) Let ( , ) G G G V E be a crisp undirected graph contain no self-loop and parallel edges. Also, let ( , , )        s. t., : [0,1]V   , : [0,1]V   and : [0,1]V   be the positive membership degree, neutral membership degree and negative membership degree respectively on the PFS V . We consider ( , , )        s. t., : [0,1]V V    , : [0,1]V V    and : [0,1]V V    as the positive membership degree, neutral membership degree and negative membership degree respectively, in the symmetric PFR E V V  . Then ( , , )V   , is said to be a PDFG if 1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) a b ab a b a b                  , ( ) G ab E  ; 2. ( ) ( ) ( ) ( ) ( ) ( ) ( ) a b ab a b a b                  , ( ) G ab E  ; 3. ( ) ( ) 2 ( ) ( ) ( ) 1 ( ) ( ) a b a b ab a b                   , ( ) G ab E  . 3. Some Operation on PDFG’s Union The union of two PDFG's ( , , )V   and ( , , )V   of the graphs ( , ) G G G V E  and ( , ) H H H V E  respectively, is denoted by  and is defined as ( , , ) G H V V       , where ( , , )                   and ( , , )                  s. t., ( )( )      ( ), if G H V V       ( ), if H G V V       A Study on Picture Dombi Fuzzy Graph 123 ( ) ( ) , if ( ) ( ) ( ) ( ) G H V V                         ( )( )      ( ), if G H V V       ( ), if H G V V       ( ) ( ) , if . ( ) ( ) ( ) ( ) G H V V                         ( )( )      ( ), if G H V V       ( ), if H G V V       ( ) ( ) 2 ( ) ( ) , if 1 ( ) ( ) G H V V                          . ( )( )ab     ( ), if ( ) G H ab ab E E     ( ), if ( ) H G ab ab E E     ( ) ( ) , if ( ) ( ) ( ) ( ) G H ab ab E E ab ab ab ab                   ( )( )ab     ( ), if ( ) G H ab ab E E     ( ), if ( ) H G ab ab E E     ( ) ( ) , if ( ) ( ) ( ) ( ) ( ) G H ab ab ab E E ab ab ab ab                  ( )( )ab     ( ), if ( ) G H ab ab E E     ( ), if ( ) H G ab ab E E     ( ) ( ) 2 ( ) ( ) , if ( ) 1 ( ) ( ) G H ab ab ab ab ab E E ab ab                   . Example 1: We consider two PDFG's ( , ) A A A    (Shown in Fig. 1(a) ) and ( , ) B B B    (Shown in Fig. 1(b) )of the graphs ( , ) A A A V E  and ( , ) B B B V E  respectively, where { , , } A V x y z , { , , } A E xy yz zx , { , , } B V y z w and { , , } B E yz yw zw . Then the union of A and B are shown in Figure 1(c). Mohanta et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 119-130 124 Figure 1(a). PDFG A Figure 1(b). PDFG B Figure 1(c). PDFG A B Join The join of two PDFG's ( , , )V   and ( , , )V   of the graphs ( , ) G G G V E  and ( , ) H H H V E  respectively, is denoted by  and is defined as ( , , , ) G H V V E       , where ( , , )                   , ( , , )                   , G H V V   , G H E E E E   (E  set of all edges joining the nodes of G V and ) H V s. t., ( )( ) ( )( ), if G H V V                ( )( ) ( )( ), if G H V V                ( )( ) ( )( ), if G H V V                ( )( )ab     ( )( ), if ( ) G H ab ab E E        ( ) ( ) , if ( ) ( ) ( ) ( ) ( ) a b ab E a b a b                 ( )( )ab     ( )( ), if ( ) G H ab ab E E        ( ) ( ) , if ( ) . ( ) ( ) ( ) ( ) a b ab E a b a b                 ( )( )ab     ( )( ), if ( ) G H ab ab E E        A Study on Picture Dombi Fuzzy Graph 125 ( ) ( ) 2 ( ) ( ) , if ( ) 1 ( ) ( ) a b a b ab E a b                  Theorem 1: The Join of two PDFG's is a PDFG. Composition The composition of two PDFG's ( , , )V   and ( , , )V   of the graphs ( , ) G G G V E  and ( , ) H H H V E  respectively, is denoted by and is defined as ( , , , ) G H V V E     , where ( , , )               , ( , , )               and 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 1 {(( , )( , )) : , ( ) } {(( , )( , )) : ( ) , } {(( , )( , )) : ( ) , } G H G H G E s t s t s V t t E s t s t s s E t V s t s t s s E t t          s. t., ( )( , )      ( ) ( ) ( ) ( ) ( ) ( )                      ( )( , )      ( ) ( ) ( ) ( ) ( ) ( )                      ( )( , )      ( ) ( ) 2 ( ) ( ) 1 ( ) ( )                       V  and ( , ) E   , ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                V  and ( , ) E   , ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                ( , ) G E   , and H V   , ( )(( , )( , ))          ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )                                               ( )(( , )( , ))         ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )                                                Mohanta et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 119-130 126 ( )(( , )( , ))         ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 4 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( )                                                                                        Example 2: We consider two PDFG's ( , ) A A A    and ( , ) B B B    of the graphs ( , ) A A A V E  and ( , ) B B B V E  respectively, where { , , } A V x y z , { , } A E xy yz , { , } B V a b and { } B E ab . Then the composition of A and B are shown in Fig. 2(c). Figure 2(a). PDFG A Figure 2(b). PDFG B Figure 2(c). PDFG A B Cartesian Product The cartesian product of two PDFG's ( , , )V   and ( , , )V   of the graphs ( , ) G G G V E  and ( , ) H H H V E  respectively, is denoted by  and is defined as ( , , ) G H V V       , where ( , , )                   and ( , , )                   s. t., ( , ) V V    , ( ) ( ) ( )( , ) ( ) ( ) ( ) ( )                             ( ) ( ) ( )( , ) ( ) ( ) ( ) ( )                             ( ) ( ) 2 ( ) ( ) ( )( , ) 1 ( ) ( )                              V  and ( , ) E   , A Study on Picture Dombi Fuzzy Graph 127 ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                V  and ( , ) E   , ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                ( , )( , ) ( )V V E       , ( )(( , )( , )) 0          , ( )(( , )( , )) 0          , ( )(( , )( , )) 0          . Remark 1: The cartesian product of two PDFG's is not necessarily a DFG. Complement of a PDFG Let ( , , ) G V   be a PDFG of the graph ( , ) G G G V E . Then the complement of is represented as ( , , ) c c c G V   and is defined as follows: c      , c      and c      ( ) c ab  ( ) ( ) , if ( ) 0 ( ) ( ) ( ) ( ) a b ab a b a b                   ( ) ( ) ( ), if 0 ( ) 1 ( ) ( ) ( ) ( ) a b ab ab a b a b                       ( ) c ab  ( ) ( ) ( ) , if ( ) 0 ( ) ( ) ( ) b b a ab a b a                   ( ) ( ) ( ) ( ), if 0 ( ) 1 ( ) ( ) ( ) b a b ab ab a b a                       ( ) c ab   ( ) ( ) 2 ( ) ( ) , if ( ) 0 1 ( ) ( ) a b a b ab a b                   ( ) ( ) 2 ( ) ( ) ( ) , if 0 ( ) 1 1 ( ) ( ) a b a b ab ab a b                        Mohanta et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 119-130 128 Example 3: We consider the PDFG ( , ) A A    of the graph ( , ) G G G V E  where { , , } G V x y z , { } G E yz . Then complement c of shown in Fig. 3(a) and Fig. 3(b) respectively. Figure 3(a). PDFG Figure 3(b). c Theorem 2: Let ( , , ) G V   be a PDFG of the graph ( , ) G G G V E . Then ( ) c c  . Homomorphism, Isomorphism, Weak isomorphism, Co-weak isomorphism Let us consider two PDFG's ( , , )V   and ( , , )V   of the graphs ( , ) G G G V E  and ( , ) H H H V E  , where ( , , )        , ( , , )        , ( , , )        and ( , , )        . (Homomorphism) A mapping :  is said to be a homomorphism, if G V  ( ) ( ( ))        , ( ) ( ( ))        and ( ) ( ( ))        ; ( ) G ab E  ( ) ( ( ))ab ab      , ( ) ( ( ))ab ab      and ( ) ( ( ))ab ab      . (Isomorphism) A mapping :  is said to be an isomorphism, if G V  ( ) ( ( ))        , ( ) ( ( ))        and ( ) ( ( ))        ; ( ) G ab E  ( ) ( ( ))ab ab      , ( ) ( ( ))ab ab      and ( ) ( ( ))ab ab      . If and are isomorphism, then we write  . (Weak-isomorphism) A mapping :  is said to be a weak isomorphism, if  homomorphism; G V  ( ) ( ( ))        , ( ) ( ( ))        and ( ) ( ( ))        . (Co-weak isomorphism) A mapping :  is said to be a co-weak isomorphism, if  is a homomorphism; ( ) G ab E  ( ) ( ( ))ab ab      , ( ) ( ( ))ab ab      and ( ) ( ( ))ab ab      . Self-complementary Let ( , , ) G V   be a PDFG of the graph ( , ) G G G V E . Then is said to be self- complementary if c . A Study on Picture Dombi Fuzzy Graph 129 Theorem 3: Let ( , , ) G V   be a self-complementary PDFG of the graph ( , ) G G G V E . Then 0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( )1 ( ) , 2 ( ) ( ) ( ) ( )s t s t s t s t s t s t                      0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( )1 ( ) 2 ( ) ( ) ( ) ( )s t s t s t s t s t s t                      0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( ) 2 ( ) ( )1 ( ) . 2 1 ( ) ( )s t s t s t s t s t s t                       Proof: Let be a self-complementary graph. So,  an isomorphism : c   s. t., G V  ( ) ( ( )) c         , ( ) ( ( )) c         ( ) G ab E  , ( ) ( ( )) c ab ab      , ( ) ( ( )) c ab ab      and ( ) ( ( )) c ab ab      . Now, we know that, ( ( )) ( ( )) ( ( ) ( )) ( ( ) ( )) ( ( )) ( ( )) ( ( )) ( ( )) c c c c c c c a b a b a b a b a b                               Or, ( ) ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( ) a b ab a b a b a b                       Or, ( ) ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( )a b a b a b a b ab a b a b a b                             Or, ( ) ( ) 2 ( )) ( ) ( ) ( ) ( )a b a b a b ab a b a b                      Or, ( ) ( )1 ( )) 2 ( ) ( ) ( ) ( )a b a b a b ab a b a b                      . In similar way we can proof the remaining two results. This completes the proof. 4. Conclusion In this paper, we have introduced the new concept of Picture dombi fuzzy graph. We have proposed some operators of union, join, composition and cartesian product of any two dombi picture fuzzy graphs and investigate many interesting properties of Dombi picture fuzzy graph. Finally, we define the complement Picture dombi fuzzy graph and the isomorphic properties on it. The concept of picture dombi fuzzy graphs can be used to model in several areas of expert systems, transportation, artificial neural networks, pattern recognition and computer networks. Author Contributions: Each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflicts of interest. Mohanta et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 119-130 130 References Alsina, C., Trillas, E., & Valverde, L. (1983). On some logical connectives for fuzzy sets theory. Journal of Mathematical Analysis and Applications. https://doi.org/10.1016/0022-247X(83)90216-0 Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems. https://doi.org/10.1016/S0165-0114(86)80034-3 Cuong, B. C., & Kreinovich, V. (2014). Picture fuzzy sets - A new concept for computational intelligence problems. 2013 3rd World Congress on Information and Communication Technologies, WICT 2013. https://doi.org/10.1109/WICT.2013.7113099 Dogra, S. (2015). Different types of product of fuzzy graphs. Progress in Nonlinear Dynamics and Chaos, 3(1), 41–56. Dubois, D., Ostasiewicz, W., & Prade, H. (2000). Fuzzy Sets: History and Basic Notions. https://doi.org/10.1007/978-1-4615-4429-6_2 Hamacher, H. (1978). Uber logische verknupfungen unscharfer aussagen und deren zugehörige bewertungsfunktionen. Progress in Cybernetics and Systems Research, 3. Havare, Özge Çolakoğlu, and H. M. (n.d.). On Corona Product of Two Fuzzy Graphs. Kaufmann, A. (1975). Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (fuzzy sets theory). Menger, K. (1942). Statistical metrics. Proceedings of the National Academy of Sciences of the United States of America, 28(12), 535. Rosenfeld, A. (1975). FUZZY GRAPHS††The support of the Office of Computing Activities, National Science Foundation, under Grant GJ-32258X, is gratefully acknowledged, as is the help of Shelly Rowe in preparing this paper. In Fuzzy Sets and their Applications to Cognitive and Decision Processes. https://doi.org/10.1016/b978- 0-12-775260-0.50008-6 Schweizer, Berthold, and A. S. (2011). Probabilistic metric spaces. Courier Corporation. Zadeh, L. A. (1965). Fuzzy sets. Information and Control. https://doi.org/10.1016/S0019-9958(65)90241-X © 2018 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).