Extension of competition graphs under complex fuzzy environment

A complex fuzzy set (CFS) is a remarkable generalization of the fuzzy set in which membership function is restricted to take the values from the unit circle in the complex plane. A CFS is an efficient model to deal with uncertainties of human judgement in more comprehensive and logical way due to the presence of phase term. In this research article, we introduce the concept of competition graphs under complex fuzzy environment. Further, we present complex fuzzy k-competition graphs and p-competition complex fuzzy graphs. Moreover, we consider m-step complex fuzzy competition graphs, complex fuzzy neighborhood graphs (CFNGs), complex fuzzy economic competition graphs (CFECGs) and m-step complex fuzzy economic competition graphs with interesting properties. In addition, we describe an application in ecosystem of our proposed model. We also provide comparison of proposed competition graphs with existing graphs.


Introduction
In 1968, Cohen [6] introduced the conception of competition graphs (CGs) to determine the problems of ecosystem. CGs have many utilizations in distinct areas of life. The analogy of Cohen was based on the fact that there is a competition between two species if both species have a common prey. Many variations of CGs are present in the literature, namely competition common enemy graph of digraph [22], p-competition graphs of digraph [12,13], competition hypergraphs [23] and tolerance competition graphs [4]. Another worthwhile generalization of competition graphs was given by Cho et al. [5] in 2000. All the introduced competi-tion graphs are crisp graphs which cannot describe all the real-world competitions. In ecosystem, the species may be vegetarian, non-vegetarian, strong, weak, and similarly the prey may be harmful, digestive, energetic, etc.
In 1965, Zadeh [25] developed the conception of fuzzy set to represent the uncertainty of human judgment by initiating the membership function which is restricted to take the values form the unit interval [0,1]. For example, today is sunny, it might be 100% true if there are no clouds, 80% true if there are few clouds, 50% true if it is hazy and 0% true if it rains all day. A graph is an easy way to interpret the information which involve the relationship between objects. Fuzzy graphs are designed to represent the structures of relationships between objects such that the existence of a concrete object (vertex) and the relationship between two objects (edge) are matters of degree. In 1975, Rosenfeld [17] initiated the notion of fuzzy graph (FG). Certain notions of FGs have been discussed in [1,8,15,27]. After the introduction of FG, many researchers turned their attention to find the competition among entities under fuzzy environment. Fuzzy k-competition and p-competition graphs were developed by Samanta and Pal [20]. Further, Samanta et al. [19] initiated m-step fuzzy competition graphs. Moreover, Habib et al. [9] initiated the concept of q-rung orthopair FCGs with their utilization in soil ecosystem. Sarwar et al. [21] introduced a new decision-making approach with fuzzy competition hypergraphs. Akram and Luqman [2] presented a remarkable contribution on fuzzy hypergraphs. Most of the real-world competitions can be represented through these FCGs but there exist some competitions in which the entities possess the periodic or two-dimensional information which cannot be represented through these FCGs. For example, in a food web, the species may be strong and vegetarian and the prey may be harmful and energetic at the same time interval. Similarly, the species may be strong or weak and the prey may be harmful or energetic under some specific time interval. This information about the species and preys is two dimensional which cannot be modeled through FCGs.
Fuzzy set (FS) theory is useful model to deal with uncertainties arising in various fields of life. But in some situations, this theory cannot model the incomplete and imprecise information of two-dimensional or periodic nature encountered in the real world. To cope with this problem, Ramot et al. [16] proposed complex fuzzy set (CFS), as a worthwhile generalization of FS in which the membership grade is restricted to take the values form the unit circle in the complex plane, and is of the form re iθ , where r is the amplitude term and θ is the phase term. The phase term θ of CFS has a great importance because it can handle the recurring problems or periodic problems phenomena with more perfection. The presence of phase term in CFS ensures that there may exist some situations where the second dimension is required. This term differentiates CFS from all other available sets of the literature. Possible application which demonstrates the new concept involve a CF representation of solar activity, signal processing application and in time series forecasting applications. Later on, Thirunavukarasu [24] introduced the idea of complex fuzzy graph (CFG). Recently, Luqman et al. [14] investigated the novel concepts of CF hypergraphs. Akram et al. [3] considered imperfect competition models in economic market structure with q-rung picture fuzzy information. Further, Garg and Rani [10,11] investigated many decision-making techniques. The motivation to this article can be described as follows.
1. The CFS, a worthwhile generalization of FS, is a proficient model to handle all the periodic or two-dimensional information due to the presence of an additional term known as phase term. This set has an edge over the other existing models of the literature due to its additional features of handling the periodic and two-dimensional information in a single set. On the other hand, if we remove the phase term of CFS then the CFS is a conventional FS with real-valued membership. 2. The competition graphs developed under CF environment are useful enough to tackle all the competitions of real world which possess the information of twodimensional nature.
In this research article, we present the innovative concept of CFCGs. We then investigate the two generalizations of CFCGs, namely CF k-competition and p-competition CFGs. Further, we investigate CFNGs, m-step CFCGs, CFECGs and m-step CFECGs. Moreover, we discuss an application of CFCGs in ecosystem to emphasize the superiority of these graphs in real life. The main contribution to this article is summarized as follows.
1. Competition graphs with its remarkable generalization are developed in CF environment to overcome the deficiencies of other existing competition graphs of the literature. 2. An algorithm is initiated to find the competition among the real-world entities with an application in ecology. 3. A comparative analysis is provided to check the superiority and authenticity of our proposed competition graphs.
We arrange this paper as follows: the next section includes some elemental definitions which are necessary for the evolution of CFCGs, followed by which we introduce the concept of CFCGs with its two remarkable extensions, namely CF k-competition graphs and p-competition CFGs. In the subsequent section, we develop the concept of CFNGs with some related theorems and then we discuss m-step CFCGs. After this, we initiate a new concept called CFECGs. The penultimate section includes an application of CFCG to reveal its importance in real life. Before the concluding section, we provide comparison of the proposed competition graphs with existing graphs. Finally, we give the conclusion.

Preliminaries
Definition 1 [20] The fuzzy competition graph (FCG) C( − → ) is an undirected graph = (W , P, Q) with same node set as in − → and have an edge connecting two distinct vertices g and f in C (  − → ) if and only if ℵ + (g) ∩ ℵ + ( f ) = ∅ in − → , and for the edge (1) Definition 2 [16] Let Z be a universal set. Then complex fuzzy set (CFS) B on Z can be given as where ξ B (q)e iϕ B (q) is the MV of the element q such that ξ B (q) ∈ [0, 1] and ϕ B (q) ∈ [0, 2π ], and i = √ −1. For every q ∈ Z , ξ B (q) and ϕ B (q) are the amplitude term and phase term, respectively. Definition 3 [7] A complex fuzzy relation (CFR) R on a universal set W × Z , can be given as are the amplitude term and the phase term, respectively. Definition 4 [26] Let B 1 and B 2 be two CFSs on a universal set Z , where Then the CF intersection of B 1 and B 2 , symbolized by B 1 ∩ B 2 , is given by the function where Definition 5 [26] Let B 1 and B 2 be two CFSs defined on a universe Z , where Then the CF product of B 1 and B 2 , symbolized by B 1 • B 2 , is given by the function where where Definition 6 [24] A fuzzy graph (FG) = (W , P, Q), on a non-empty set W is said to be a complex fuzzy graph (CFG), where P and Q are CFS and CFR on W such that ξ Q (g, f ) ≤ (ξ P (g) ∧ ξ P ( f )), (for amplitude term) (12) ϕ Q (g, f ) ≤ (ϕ P (g) ∧ ξ P ( f )), (for phase term) (13) for all g, f ∈ W .
for all g, f ∈ W .
The list of acronyms in research paper is given in Table 1.

Complex fuzzy set applied to competition graphs
This section presents some basic definitions which are necessary to define for the evolution CFCGs.

Definition 8 For a node g of a CFDG
the CF out-neighborhood (CFON) is a CFS, given by

Definition 9
For a node g of a CFDG − → = W , P, − → Q , the CF in-neighborhood (CFIN) is a CFS, given by ℵ n (g) = (S n g , m n g e iη n g ), The CFDG is shown in Fig. 1. CFON and CFIN of the nodes are given in Table 2.
same CF node set as in − → and possesses a CF edge in C − → joining two distinct nodes g and f if and only if the CFS Fig. 2a, given by The CFON of the nodes are shown in Table 3. Table 4.
The corresponding CFCG is shown in Fig. 2b.
Then the cardinality of CFS B is a positive complex number symbolized by |B| ξ e i|B| ϕ , is defined as for all q ∈ Z .

Definition 13
Suppose k = re iθ be a complex number and is an undirected CFG = (W , P, Q) with same CF node set as in − → and have a CF edge in C k ( − → ) connecting two distinct vertices g and f

Fig. 2 Complex fuzzy digraph and complex fuzzy competition graph
The CFON of the nodes are given in Table 5. Next, we have  Fig. 4.

Definition 14 Consider a CFG
then the edge (g, f ) is said to be a strong edge.
The the support of CFS is a subset B 0 of Z , defined as Now, we discuss p-competition complex fuzzy graph.

Definition 16 Letp be positive integer. Thep-competition
with same CF node set as − → and possesses a CF edge joining two distinct nodes g and The CFON of the nodes are give in Table 6. The corresponding two-competition CFG has only one edge as Fig. 5b.
Then by Definition 16, we have

Complex fuzzy set applied to neighborhood graphs
Now, we discuss complex fuzzy neighborhood graphs by the combination of CFS and neighborhood graphs. Before the introduction of CF neighborhood graphs (CFNGs), we first define complex fuzzy open and closed neighborhood of a node in CFGs below.
For each node g ∈ W , CF singleton set is given as where ξ P : {g} → [0, 1] and ϕ P : {g} → [0, 2π ] given by ξ P (g) = ξ P (g) and ϕ P (g) = ϕ P (g), respectively. Now, we define CF closed neighborhood (C FC l N ) of a node g as Next, we define CF open neighborhood graphs and CF closed neighborhood graphs. Then on the bases of these CFGs, we define complex fuzzy k-neighborhood graphs of both close and open types.
and possesses a CF edge joining two distinct nodes g and Definition 19 Let = W , P, Q be a CFG. Complex fuzzy closed neighborhood graph (C FC l N G) of is a CFG denoted by N[ ] = W , P, Q with same CF node set as and possesses a CF edge joining two distinct nodes g and Definition 20 Complex fuzzy (k)-neighborhood graph of a CFG = (W , P, Q) is a CFG N k ( ) = (W , P, Q ) with same node set as and possesses a CF edge joining two distinct nodes g and f in N k ( ) if and only if x > r and ω > θ and the MV of the edge (g, f ) in N k ( ) is calculated as Here, k is a complex number given by k = re iθ and |ℵ(g) ∩ ℵ( f )| = x e iω .
The CFDG is displayed in Fig. 6a. The CF neighborhoods of the nodes are presented in Table 7.
The height and cardinality of CFSs (ℵ(g) ∩ ℵ( f )) and Table 8. The Thus, for each edge (g, f ) in there exists an edge (g, f ) in N[ ].

Definition 22 Let
is defined as U( ) = (W , P, Q), where P and Q are CFS and CFR on W such that for all g, f ∈ W . Now, we establish a relation between N k ( ) and C k ( ). Also, let C k ( − → ) = W , P,C and N k (U( )) = W , P,C . The complex fuzzy node of C k ( ) is same as − → . Also the node set of an underlying CFG is same as the complex fuzzy node of CFDG. Hence, N k (U( )) has the same complex fuzzy node set as C k ( ). Now, we need to show that complex fuzzy edge set of C k − → is equal to the edge set of N k (U( )) . For this, we need to show that ξC(g, f ) = ξC (g, f ) and ϕC(g, f ) = ϕC (g, f ), for all g, f ∈ W .

Theorem 4 If a symmetric CFDG
Case 2 If r > r and θ > θ in − → , then ξC(g, f ) > 0 and ϕC(g, f ) > 0 in C k − → . Then amplitude and phase term in C k − → are given by As − → is symmetric CFDG, x > r and ω > θ in U( ). The amplitude and phase term of MV in N k (U( )) are As − → is symmetric then it is clear thath Hence, ξC(g, f ) = ξC (g, f ) and ϕC(g, f ) = ϕC (g, f ) for all g, f ∈ W .

Complex fuzzy set applied to m-step competition graphs
Now, we define m-step CFCG by combining CFS with m-step CGs. Before defining m-step CFCG , we first discuss m-step complex fuzzy digraph (m-step CFDG), complex fuzzy mstep out-neighborhood of a node and then complex fuzzy m-step in-neighborhood of a node below.

Definition 23 Let
Example 6 Let − → = W , P, − → Q be a CFDG as presented in Fig. 7, given by In Fig. 7a, two-step out-neighborhood of the node a is the CFS ℵ for all g, f ∈ W . Fig. 7a, given by The two-step out-neighborhood of the nodes are displayed in Table 9. Table 9 Two-step out-neighborhood of the nodes The two-step CFCG of − → is shown in Fig. 7b. Now, we define the strength of CFCG below.

Definition 27
Let − → = W , P, − → Q be a CFDG. Let the common node of m-step out-neighborhoods of nodes For the prey x, the strength is measured by the mapping s : W → {z|z ∈ C : |z| ≤ 1} such that

Example 8
Let − → = W , P, − → Q be a CFDG as shown in In Fig. 8, the strength of the prey f is Hence, the prey f is two-step strong node as 0.6 > 1 2 and 1.1π > 1π.

Theorem 6 If a node x of
− → is strong, then in strength of x, the common node of m-step out-neighborhood of nodes s 1 , s 2 , · · · , s n , i.e., there exist CFDPs This completes the result.
Then the amplitude and phase term for the edge (g, f ) is Then clearly ξC(g, f ) > 1 2 ξ P (g) ∧ ξ P ( f ) and ϕC(g, f ) > 1 2 2π . Hence, the edge (g, f ) is strong. Since all the edges are strong. Now, we establish a relation between m-step CFCG of a CFDG and CFCG of m-step CFDG. Definition 28 Complex fuzzy m-step neighborhood of a node g of a CFG = (W , P, Q) is the CFS

Definition 29 Let
for all g, f ∈ W .

Example 9
Let − → = W , P, − → Q be a CFG as presented in Fig. 9a, given by The CF two-step neighborhood of the nodes are displayed in Table 10.
The two-step CFNG of − → is shown in Fig. 9b.

Complex fuzzy economic competition graphs
Now a days, the transfer of money from one place to another place is very easy because of the advancement of internet. During this exchange of money, there are destinations and sources. Let us consider three projects, namely, C 1 , C 2 , C 3 as sources of money and let five organizations, namely D 1 , D 2 , D 3 , D 4 , D 5 in which some projects are running in (see Fig. 10). Here, the projects and organizations can be taken as vertices and interconnection between them can be illustrated by arcs. The two organizations are connected by an edge if they are under a certain project, and so on. The graph of this type is called economic competition graph. Now, we develop complex fuzzy economic competition graphs and m-step complex fuzzy economic competition graphs.

Definition 31
for all g, f ∈ W .
The CFDG is presented in Fig. 11a. The CFIN of the nodes are shown in Table 12. The heights of CFSs ℵ n (a) ∩ ℵ n (b) and ℵ n 2 (a) ∩ ℵ n 2 (b) are given in Table 13.
The corresponding CFECG and two-step CFECGs are shown in Fig. 11b and c.
Theorem 9 CFECG and CFCG of any complete graphs are same.
Also, the MV of the edge (g, As − → is a complete fuzzy digraph, then clearlȳ i.e., Then ξ Q (g, f ) = ξC(g, f ) and ϕ Q (g, f ) = ϕC(g, f ). Hence, the result is proved.  The CFONs of the nodes are displayed in Table 15. There exist only two edges in CFCG as The corresponding CFCG is shown in Fig. 13. Now, sea urchin and starfish are connected by an edge, and similarly sea urchin and snail are connected by an edge in the CFCG. There is no edge in the CFCG for the other pair of vertices. The MVs of the edges between sea urchin and starfish is 0.48e i0.45π , and sea urchin and snail is 0.49e i0.16π calculated by the definition of CFCG. Hence, sea urchin and starfish; sea urchin and snail compete for sea otter and starfish, respec- tively. Thus, there is competition between sea urchin and starfish; sea urchin and snail in the ecosystem. The method which is used in our ecosystem application is shown in Table  16 .

Comparative analysis
CGs are very useful to show the competitions among the different objects. Fuzzy competition graphs were proposed by Samanta et al. [20] to show all the competitions of the real world. Fuzzy competition graphs have larger ability to deal with vagueness and uncertainty. There are many applications of fuzzy competition graphs in various fields of life. Ecosystem is one of these applications of fuzzy competition graphs. For example, in ecology, let {( f 1 , 0.9), ( f 2 , 0.8), ( f 3 , 0.9), ( f 4 , 0.8), ( f 5 , 0.7)} be the set of species and let {(  Step 1 Assign the MVs for the set of n species in the food web. − → = ( P, − → Q )(say).
Step 2 If for any two nodes y i and y j , ξ Q (y i , y j ) > 0, then (y j , ξ Q (y i , y j )e iϕ Q (y i ,y j ) ) ∈ ℵ p (y i ).
Step 3 Find out the CFON ℵ p (y i ) for all vertices y i , y j .
Step 6 If ℵ p (y i ) ∩ ℵ p (y j ) = ∅ then draw an edge (y i , y j ).
Step 7 Repeat the Step 6 for all the disjoint nodes of the digraph.
Step 8 Calculate the MVs using 2π . tence of these species but it does not give the information that how much the specie is strong. Similarly, the MVs of the edges show only the likeness of preys for the predators but fails to illustrate that how much the prey is tasty to eat for the predator. This information about the preys and predators is two-dimensional which cannot be modeled by one dimensional phenomenon. This lack of knowledge motivates us to introduce the more generalized concept, CFCGs of FCGs, which have a capability to handle the two dimensional data with more perfection. The proposed work provides us a better platform to show the periodicity of the phenomenon. An application is designed in our presented work, in which the phase terms of the nodes and edges show the strength and tastiness of these species under some specific time interval. Therefore, CFCGs are more useful then the FCGs, as these graphs can handle the two-dimensional information.
The aptitude of these graphs to represent the two-dimensional phenomena make it worthier to manage the ambiguous and intuitive data.

Further discussion
From the above comparison, it is clear that the proposed competition graphs are more generalized and superior the the existing competition graphs because they are efficient to deal with two-dimensional data of species in a single set. The drawbacks of the existing CGs and the main advantages of proposed CGs are given below.
1. The existing competition graphs are developed under fuzzy environment in which the uncertainties and ambiguities present in the data are handled with the help of degrees of membership that are the subset of real numbers, which may lose some useful information. 2. The proposed competition graphs are developed under CF environment in which the uncertainties of data are handled by complex numbers. The CFS can handle two dimensional and periodic information more precisely then the other existing models. The phase term of CFS is efficient which discriminates this set from all other existing models of the literature. 3. On removing the phase term of CFS, the CFS reduces to ordinary FS. Thus, CFS is an effective generalization of FS.
The characteristics comparison of our proposed approach with different existing approaches is given in Table 17.

Conclusion
CFS is an indispensable tool to deal with two-dimensional or periodic information due to the existence of additional term named as phase term which discriminate this set from all other existing sets of literature. In this paper, we have introduced a novel concept of CGs and economic competition graphs under CF environment. We have also investigated two worthwhile extensions of complex fuzzy competition graphs, named as complex fuzzy k-competition graphs and p-competition complex fuzzy graph. Moreover, we have developed neighborhood graphs and m-step neighborhood graphs under complex fuzzy environment with some of their remarkable results. In addition, to reveal the importance of these competition graphs in realistic situations, we have designed an application of complex fuzzy competition graphs in ecology. Finally, we have compared our proposed complex fuzzy competition graphs with existing graphs to show the superiority and authenticity of proposed graphs which leads us to the result that competition graphs are useful to represent the competition among those entities which possess the two-dimensional information. Our next interest of research is to extend our work to (1) complex fuzzy mixed graphs,

Compliance with ethical standards
Conflict of interest The authors declare no conflict of interest.
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