https://orcid.org/0000-0003-0884-3056
Figures
Abstract
The extension of bipolar fuzzy graph is bipolar fuzzy incidence graph (BFIG) which gives the information regarding the effect of vertices on the edges. In this paper, the concept of matching in bipartite BFIG and also for BFIG is introduced. Some results and theorems of fuzzy graphs are also extended in BFIGs. The number of operations in BFIGs such as augmenting paths, matching principal numbers, relation between these principal numbers and maximum matching principal numbers are being investigated which are helpful in the selection of maximum most allied applicants for the job and also to get the maximum outcome with minimum loss (due to any controversial issues among the employees of a company). Some characteristics of maximum matching principal numbers in BFIG are explained which are helpful for solving the vertex and incidence pair fuzzy maximization problems. Lastly, obtained maximum matching principal numbers by using the matching concept to prove its applicability and effectiveness for the applications in bipartite BFIG and also for the BFIG.
Citation: Rehman FU, Rashid T, Hussain MT (2023) Applications of maximum matching by using bipolar fuzzy incidence graphs. PLoS ONE 18(8): e0285603. https://doi.org/10.1371/journal.pone.0285603
Editor: Ronnason Chinram, Prince of Songkla University, THAILAND
Received: October 28, 2022; Accepted: April 26, 2023; Published: August 22, 2023
Copyright: © 2023 Rehman et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
A graph is more suitable to explain any kind of information along with the mutual relationship between different types of objects. The relationship between different entities are represented in terms of edges while entities do represent vertices. Zadeh was the first one who introduced the theory of fuzzy sets (FSs), which provides us the grade of membership of an object [1]. This theory opened an energetic area of research in various disciplines in the fields of automata, medical sciences, computer networking, statistics, social sciences and its various subbranches and disciplines, management sciences, engineering and graph theory etc. In this way Zadeh was the one who paved the way for Rosenfeld who introduced the fuzzy graph (FG) theory [2]. Rosenfeld was the one who presented several graph theoretical ideas, for example path, cycle and connectedness. FG is used when there is an inadequacy in the explanation and justification to various objects or entities and it was of great help to researcher. Mordeson put forth the concept regarding FGs and determined basic properties of it [3].
FGs are unable to give the detailed information about the impact of vertices on edges. This shortage in FGs was the basic problem which is covered by fuzzy incidence graphs (FIGs). The concept of FIG was put forth by Dinesh [4]. Different concepts with regard to the connectivity were put forth by Moderson and Mathew [5]. They introduced various structural properties and establish the prevalence of a strong path between any of the node are pair of a FIG. Inter connectivity between index and wiener index with regard to the FIGs was put forth by Fang et al. [6].
Zhang introduced the concept of bipolar fuzzy sets (BFSs) [7]. The membership grade in the extension of FS to BFS is [-1,1]. An element has 0 grade in BFS if it has zero role on the resultant property. In such a way the membership degree of an element would be(0, 1] which will explain its properties to some extent. If membership grade of an element is [−1, 0) which tells that its marginal pleases the implicit counter property. The idea of the symbolization of bipolar fuzzy graphs (BFGs) along with the matrices in FGs, regular and irregular BFGs, hyper BFGs and antipodal BFGs along with their various applications, properties and significance was explained by Akram et al. [8–14]. Mohanta et al. gave a study of m-polar neutrosophic graph with applications [15]. Xiao et al. gave the study on regular picture fuzzy graph with applications in communication networks [16].
FG gives only positive membership values of vertices and edges whereas FIGs gives the positive membership values of vertices, edges and incidence pairs. BFG are able to give positive and negative membership values of vertices and edges. FGs, FIG and BFGs are unable to give the detailed information about the impact of vertices on edges. This shortage in BFGs was the basic problem which is covered by BFIGs. BFGs are able to give positive and negative membership values of vertices and edges whereas BFIGs are able to give positive and negative membership values of vertices, edges and incidence pairs. The concept of BFIG was put forth by Gong and Hua [17]. There are multiple reasons to introduce the concept of matching in bipartite BFIG and for BFIG. Let us consider an example to understand the concept of BFIG, if nodes reflects distinct companies and edges are the roads which connects these companies, then an BFG will give us the information of traffic between these companies. The company which have more number of employees will have the foremost infrastructure in the company. Hence, if C1 and C2 be two companies and C1C2 is a road between these companies, then (C1, C1C2) will be the incline system from the the company C1 using the road C1C2 to the company C2. Similarly, (C2, C1C2) will be the incline system from the company C2 using the road C1C2 to the company C1. Both C1 and C2 have the impact of 1 on C1C2 in un-weighted graphs. But, the impact of C1 on C1C2 will be (C1, C1C2) is 1 whereas (C2, C1C2) is 0 in a directed graph. This is the main concept of BFIG.
Matching is important area in the graph as well as in the FG theory. It was Shen and Tsai who introduced the concept of optimal graph matching approach for solving the task assignment problem [18]. The concept of matching in FGs was introduced by Ramakrishnan and Vaidyanathan [19]. Later on, Mohan and Gupta further worked and gave the Graph matching algorithm for task assignment problem [20]. Matching numbers in fuzzy graphs are explained by Khalili et al. [21]. Our first objective is to find out maximum matching principal numbers in bipartite BFIG and for BFIG which are helpful to reflect the selection maximum applicants and their maximum working with minimum loss due to some controversial issues. Besides of this, some of the characteristics of the matching as well as bounds in bipartite BFIGs and BFIG have also been discussed. By using related examples, a detailed study has been carried out in the fields of matching number for the BFIGs.
Section 2 gives some preliminary definitions which are helpful to understand the next sections of the article. Section 3 contains some definitions, examples, results and theorems related to the concept of matching in BFIG. Section 4 gives mathematical model for obtaining MMVBFIN and MMBFIN for bipartite BFIG and BFIG. Section 5 contains comparative analysis is discussed for matching in bipartite BFIG and BFIG. Lastly, conclusions and prospects are explained in section 6.
2 Bipolar fuzzy incidence graph
This segment consists of some basic definitions including FS, BFS, FG, incidence graph (IG), FIG, BFG, BFIG, complete bipolar fuzzy incidence graph (CBFIG), matching, some concepts related to matching in classical theory and some examples. In this article, V, E = V × V and I = V × E represents the set of vertices, set of edges and set of incidence pairs, respectively. Let G = (V, E) be a crisp graph. A set of pairwise non-adjacent edges is known as matching. A matching
is known to be perfect matching if it covers all the vertices of the crisp graph G and if a matching
covers maximum vertices then it is known as maximum matching. A crisp graph G is said to be nearly perfect matching if only one vertex is unmatched. The number of edges in a maximum matching is known as the matching number and is denoted by
. A track in which edges are alternating in
and
is known as
-alternating track and if neither its starting and nor its final vertex is covered by
then, it is known as
-augmented track.
Definition 2.1: [1] Let V be the FS from the universal set U is defined as V = {(χu, νv(χu)) : νv(χu) ∈ [0, 1], χu ∈ U}.
Definition 2.2: [1] Let V be the any nonempty set from the universal set U, a mapping U : V → [0, 1] is known as fuzzy subset.
Definition 2.3: [2] Let v be the fuzzy subset of the set V and E be the fuzzy subset of V × V. A FG is a pair, such that E(vi, vj) ≤ min(ν(vi), ν(vj)), ∀vi, vj ∈ V.
Definition 2.4: [7] Let U be a universal set. A BFS B on U is defined as .
Definition 2.5: [8] Let be the BFG of FG
with the given conditions:
- (a)
and
,
- (b) E ⊆ V × V,
and
, such that
,
, ∀vi, vj ∈ E.
Definition 2.6: [4] Let be an IG of a crisp graph G = (V, E). Then
be the FIG of IG
, where V*, E* and I* are the fuzzy subsets of V, V × V and V × E respectively, such that I*(vi, vivj) ≤ min(V*(vi), E*(vivj)).
In Fig 1, the members of I are (q0, q0q1), (q1, q0q1), (q1, q1q2) and (q2, q1q2).
Definition 2.7: [17] Let be the BFIG of FG
with the given conditions:
- (a)
and
,
- (b) E ⊆ V × V,
and
,
- (C) I ⊆ V×E,
and
,
such that
,
, ∀vi, vj ∈ V and vivj ∈ E.
Definition 2.8: [17] Let be the BFIG, then it is known as CBFIG if it satisfies the following conditions
,
, ∀(vi, vivj) ∈ I.
3 Matching in bipolar fuzzy incidence graph
This segment consists of some definitions like, support of BFIG, degree of vertices, degree of edges and incidence pairs in BFIGs, path, strength, strength of connectedness, matching, matching principal numbers, maximum matching principal numbers, some examples and theorems.
Definition 3.1: Let be the BFIG, then the support of BFIG is denoted by
and is defined as
,
,
.
Definition 3.2: Let be the BFIG.
- Two vertices v0 and v1 are said to be connected if there exist a path from v0 to v1 such that v0, (v0, v0v1), v0v1, (v1, v1v0), v1.
- Vertex v0 and an edge v0v1 are said to be connected if there exist a path such that v0, (v0, v0v1), v0v1 between them.
Definition 3.3: Let be the BFIG, then
- The degree of any vertex vi ∈ V* in
is defined as
.
- The degree of any edge E(vivj) ∈ E* in
is defined as deg(vivj) = ∑vk∈VE(vivk)+ ∑vk ∈ VE(vjvk) − 2E(vivj).
- The degree of any incidence pair I(vi, vivj) ∈ I* in
is defined as deg(vi, vivj) = ∑vk∈VI(vi, vivk) + ∑vk∈VI(vj, vjvk) − 2I(vi, vivj).
Example 3.4: Consider a as shown in Fig 2. We are going to calculate the degree of vertex and the incidence pair as well. The degree of distinct vertices is given as:
deg(v0) = (0.4, −0.1), deg(v1) = (1.1, −0.2), deg(v2) = (1.4, −0.3) and deg(v3) = (0.7, −0.2).
Similarly, the degree of distinct incidence pairs is given as:
deg(v0, v0v1) = (1.5, −0.3) − (0.4, −0.1) = (1.1, −0.2), deg(v1, v1v2) = (1.8, −0.4) and deg(v2, v2v3) = (1.4, −0.3).
Definition 3.5: The strength of connectedness between in the BFIG is denoted by
, where
and
are the maximum and the minimum of the strengths of all the paths between vi and vj, respectively.
Throughout this article, strength of path is denoted by S(P), strength of connectedness will be represented by I∞(vi, vivj).
is a matching of
with set of vertices, edges and incidence pair
,
and
respectively. A collection of all matchings in
is denoted by
. A matching in
is known to be covering matching if
.
Definition 3.6: Let be the BFIG and its subgraph
is known as matching in
if exactly single
can be obtained
for which v ≠ u and μM(uv) ≥ 0.
Example 3.7: Consider a BFIG as given in Fig 3 with one possible matching. In this BFIG, we have:
,
and
.
Corollary 3.8: Let be the BFIG. Any matching in
is induced by a matching in G.
Proof As a matching is taken as the set of triples like 〈…, viejvk, …〉 and we must mention the vertices and incidence pair specifically. So, a matching M as presented in Fig 3 and can be written as 〈(v0, v0v1), (v2, v2v3)〉.
Proposition 3.9: Let be the BFIG. If
is a matching in
, then I∞(vi, vivj) = I(vi, vivj), for all
.
Proof Let . If there is a path which connects vi and vj, then this path is a single incidence pair (vi, vivj) and S(P) = I(vi, vivj), otherwise we have S(P) = I(vi, vivj) = 0. So, I∞I(vi, vivj) = I(vi, vivj) for each case.
Theorem 3.10: Let be the BFIG containing a matching
, then deg(vi) = deg(vj) = (vi, vivj) and deg(vi, vivj) = 0 for every
.
Proof As , there is only one
is available such that I(vi, vivj)>0, we get
,
and
,
,
degI(vi, vivj) = I(vi, vivj) + I(vj, vjvi) − 2I(vi, vivj) = 0.
Definition 3.11: Let is a matching in BFIG
. Then,
- The matching bipolar fuzzy incidence number of
can be described as,
.
- The matching edge bipolar fuzzy incidence number of
can be described as,
.
- The matching vertex bipolar fuzzy incidence number of
can be described as,
.
- The matching crisp number of
can be described as,
.
We consider ,
and
as matching bipolar fuzzy incidence principal numbers (MBFIPNs) of
.
Example 3.12: A BFIG with a possible matching is presented in Fig 3. The MBFIPNs are obtained as,
,
and
.
Definition 3.13: Let is a matching in BFIG
. Then,
- The maximum matching bipolar fuzzy incidence number of
can be described as:
.
- The maximum matching edge bipolar fuzzy incidence number of
can be described as:
.
- The maximum matching vertex bipolar fuzzy incidence number of
can be described as:
.
- The maximum matching crisp number of
can be described as:
.
We consider ,
and
as MMBFIN, MMVBFIN and MMCN.
In classical graph theory, a lot of matchings with same MMCN can be found but in fuzzy sense, we can differentiate them in terms of fuzzy values.
Example 3.14: Consider a BFIG as given in Fig 4.
Now, we will find out all possible matchings, MBFIPNs and after that MMBFIN, MMVBFIN and MMCN for Fig 4 as presented in Table 1.
Now, it is easy to calculate the following numbers: ,
and
.
Proposition 3.15: Let be a matching in BFIG
. Then,
, we have
.
Proof Let . As
is a BFIG,
I(vi, vivj) ≤ V(vi) ∧ E(vivj) and E(vivj) ≤ V(vi) ∧ V(vj) for all . So, we have:
.
Definition 3.16: Let be a BFIG containing a matching
. A bipolar fuzzy
-augmenting track in
is an
-alternating track containing different nodes vo, v1, v2, …vn, vn+1. So, as a result:
- I(vi−1, vi−1vi) > 0, where i = 1, 2, 3, …, n, n + 1,
,
- Neither v0 nor vn+1 are in
.
Corollary 3.17: Let be a BFIG containing a bipolar fuzzy incidence
-augmenting track P. Then, it is
-augmenting track in crisp graph G = (V, E).
Proof Let be a BFIG containing a matching
. P be a bipolar fuzzy incidence
-augmenting track and their symmetric difference is denoted by ⊕. As
represents a collection of nonadjacent incidence pairs and I(vi, vivj) > 0 for all
, which shows that
is a matching.
Theorem 3.18: Let be a BFIG containing a matching
. If P is a bipolar fuzzy incidence
-augmenting track then,
.
Proof Let P be a bipolar fuzzy -augmenting track, by using definition 3.16, we have
.
Now, by using definition 3.11, we have:
,
,
As a result, we get: .
Example 3.19: Consider a BFIG as given in Fig 5. Now, matching
and the incidence pair in the augmented track between v1, v5 is 〈(v2, v2v3)〉. But in
, it is 〈(v1, v1v2), (v3, v3v5)〉 as presented in Table 2 for Fig 5.
Theorem 3.20: Let be a BFIG containing a matching
with MMVBFIN. Then,
has MMCN.
Proof Let be a BFIG containing a matching
with MMVBFIN. It is adequate to prove that
has foremost number of incidence pairs. As
is a matching in crisp graph G. Now, if there exist any
-augmenting track P, then by using the symmetric difference
increases the MVBFIN by using theorem 3.18. So, the condition of maximum incidence pairs holds. Hence, in a matching M with MMVBFIN there exist MMCN. Therefore, according to the Berge’s theorem,
has the foremost number of edges, if there is no
-augmenting track.
Remark 3.21: As a bipolar fuzzy incidence covering matching (BFICM) includes all the vertices of BFIG, hence each bipolar fuzzy covering matching acknowledges MMVBFIN.
Corollary 3.22: Let be a BFIG containing BFICM
, then
admits MMVBFIN.
Proof Let be the BFIG. If there does not exist any
-augmenting track then, there must exist at least one
,. So, according to the Berge’s theorem, if there does not exist any
-augmenting track then,
has the foremost number of edges. Hence, it must admits MMVBFIN.
Definition 3.23: Let be the BFIG. Then:
- Consider two arbitrary nodes v1, v2 ∈ V*. v1 is known as bipolar fuzzy incidence prior to v2 if and only if
,
and l(v1) ≤ l(v2). It is denoted by v1 ≺ v2.
- Let Let
be the BFIG with two matchings
and
, for which
. Then,
is known as bipolar fuzzy incidence prior to
if and only if
.
- Consider
is the set which includes all the possible matchings in
with MMCN. A matching
is known as bipolar fuzzy incidence strong vertex matching, if
, where i = 1, …, n.
Proposition 3.24: Let be a BFIG. If
is a bipolar fuzzy incidence strong vertex subgraph in
, then
.
Proof Consider from
. By using the theorem 3.20, any
admits the MMCN. So, by using the definition of bipolar fuzzy incidence strong vertex matching, we have
. Hence,
.
Definition 3.25: Let be the BFIG. Then, it is called bipartite bipolar fuzzy incidence graph (BBFIG) if the set of vertices V can be divided into two subsets V1 and V2 such that each edge either connects a vertex from V1 to V2 or a vertex from V2 to V1.
Remark 3.26: Every perfect matching of is the spanning graph of
.
Now, we are going to construct pseudo-fuzzy restrictions for the BBFIG which will be used in different methods for finding the matchings with MBFIPNs.
Definition 3.27: Let be the BBFIG. The set of vertices V is divided into two subsets V1 and V2 such that V = V1 ∪ V2. We consider the pseudo bipolar fuzzy incidence restrictions for
as:
,
,
and
.
In the same way, we consider the pseudo bipolar fuzzy incidence restrictions for as:
,
,
and
.
Theorem 3.28: Let and
be the two matchings, respectively in the pseudo bipolar fuzzy incidence restrictions
and
of BBFIG
with V = V1 ∪ V2 as the set of vertices. Then, there is a new matching
, which matches all the vertices covered by
and
.
Proof Let be the BBFIG. Let A ⊆ V1 and B ⊆ V2. If
has a matching covering A and matching covering B, then it has a matching covering A ∪ B. Hence, if
and
be the two matchings, respectively in the pseudo fuzzy restrictions
and
of BBFIG
with V = V1 ∪ V2 as the set of vertices. There is a new matching
, which matches all the vertices covered by
and
.
4 Mathematical model
In this section, we will discuss the method for obtaining MMBFIN. There are two objectives for achieving MMBFIN. First is to give the maximum jobs to the applicants where two factors are focused: (1) maximize positive membership value which reflects their maximum working efficiency of the applicants. (2) minimizing the negative membership value which reflects the bad performance due to controversial issues among them. Second objective is to maximize the working of the employees of a company. To achieve the first objective BBFIG is used whereas to achieve the second objective BFIG is used.
4.1 MMVBFIN problem in BBFIG
In this part, we are explaining the process to obtain the MMVBFIN in BBFIG. In this process the main points are;
Step—1 Arrange the vertices of V1 and V2 in ascending order.
Step—2 Let v1 ∈ V1 be a vertex having highest membership value which in matched with vertex from V2 having highest membership value and obtain the matching for the graph.
Step—3 Consider that matching and find another matching by taking symmetric difference of . Continue this process till there is no augmenting path or obtain the matching which is already obtained.
Step—4 Choose the strongest matching and obtain the MMVBFIN.
Example 4.1: Let be the BBFIG as presented in Fig 6.
Step—1 by using this step we have arranged the vertices according to their membership values and presented in Fig 7.
Step—2 By using second step, we get which is
and its augmenting path is represented by (u1, u1v2), (u2, u2v1) as shown in Fig 8.
Step—3 By using third step, we get which is
and its augmenting path is represented by (u3, u3v0). Now, obtaining again its augmenting path presented in Fig 9;
Step—3 By using third step, we get which is
and its augmenting path is represented by (u1, u1v2), (u2, u2v1) which is same as
as shown in Fig 10.
Hence, by using step-4, we get as our final matching and MMVBFIN = (2.7, −2.1).
4.2 MMVBFIN problem in arbitrary BFIG
In this part, we are explaining the process to obtain the MMVBFIN in the arbitrary BFIG. In this process the main points are;
Step—1 Arrange the vertices such that v1 is strongest vertex. v2 is the vertex which is connected with v1 and weaker than v1.
Step—2 Consider an incidence pair, except (v1, v1v2) connected with v2 which is our first matching. If, no such incidence pair is found then, start from (v1, v1v2).
Step—3 Obtain the strong vertex augmenting path from and continue this process un-till there is no augmenting path.
Step—4 Choose the maximum vertex matching and obtain MMVBFIN.
Application Let be the BFIG as shown in Fig 11.
Step–1 By using step one, v1 is strongest vertex. v2 is the vertex which is connected with v1 and weaker than v1. v3 is weaker then v2 and so on.
Step—2 The incidence pair is our first matching and the augmenting path is p1 = 〈(v2, v2v1), (v3, v3v4)〉.
Step—3 Taking is second matching and the augmenting path is p2 = 〈(v1, v1v5), (v4, v4v6), (v2, v2v3)〉.
Continuing this process, is third matching and the augmenting path is p3 = 〈(v2, v2v1), (v5, v5v6), (v3, v3v4)〉.
Furthermore, is our last matching and its augmenting path is again same as
.
Step—4 Now, M3 and M4 both have MMVBFIN. is maximum vertex matching and MMVBFIN = (4.2, −1.8) for Fig 11.
4.3 MMBFIN problem in BBFIG
The matching concept is used for BBFIG because it is considerably more complicated in BFIG. The question is weather the difficulty is reducible or not? By using the proposed method for the BFIG, the answer is “yes”. Firstly, we are going to introduce the method for obtaining the MMBFIN in the BBFIG. The greatest membership value of the incidence pair in is presented by
. In simple words:
and
.
Let be the BBFIG with V = V1 ∪ V2. For any vertex v ∈ V, there exist one and only one adjacent incidence pair of v which is present in the matching process. Taking y : I → {0, 1} as an incidence vector which reflects the presence or absence of an incidence pair in the matching. The MMBFIN problem in the BBFIG can be described as:
z = (max∑y(I)Ip, min∑y(I)In)
subject to: ∑I = (vi, vivj)I = 1, ∀v ∈ V, I ∈ {0, 1}.
Example 4.2: Let be the BBFIG with
. Consider there are three jobs and four applicants which are represented by {u1, u2, u3} and {v0, v1, v2, v3}. Our goal is to assign jobs to applicants and every job is assigned to at most one applicant such that maximum number of jobs will be filled by using the matching concept.
z = (0.3(u2, u2v0) + 0.3(u2, u2v1) + 0.2(u3, u3v0) + 0.2(u3, u3v2) + 0.5(u1, u1v1) + 0.5(u1, u1v2) + 0.5(u1, u1v3), − 0.4(u2, u2v0) − 0.4(u2, u2v1) − 0.7(u3, u3v0) − 0.7(u3, u3v2) − 0.1(u1, u1v1) − 0.1(u1, u1v2) − 0.1(u1, u1v3)),
Subject to:
(u2, u2v0) + (u2, u2v1) = 1,
(v0, v0u2) + (v0, vou3) = 1,
(u3, u3v0) + (u3, u3v2) = 1,
(v1, v1u2) + (v1, v1u1) = 1,
(u1, u1v1) + (u1, u1v2) + (u1, u1v3) = 1.
(v2, v2u3) + (v2, v2u1) = 1,
(v3, v3u1) = 1,
The above system of linear equations is solved by using the simplex method and got for Fig 12.
By using the matching process {v0, v2, v3} got the job. For the job u1 the applicant v3 is selected because the negative membership value of v3 is less then v1, v2. Similarly, other applicants are selected. By using this concept maximum applicants are selected as all the vacancies are filled.
4.4 MMBFIN problem in arbitrary BFIG
In this part, we are explaining the process to obtain the MMBFIN in the arbitrary BFIG. In this process the main points are;
Step—1 Let the matching be empty by default for
.
Step—2 The incidence pair I is known as elected incidence pair for the subgraph Hi. The criteria of selecting the vertices for obtaining the MMBFIN of is
and
. If there are more then one incidence pairs which fulfills this criteria then select the incidence pair having maximum membership value.
Step—3 Select the alternating incidence pair from adjacent incidence pairs. If there are more then one incidence pairs then, make a matching as for every incidence pair.
Step—4 Repeat the same process for all the possible of
.
Step—5 Select the matching having which gives MMBFIN and also shows the maximum working of the employees in a company.
Application Consider a department have 6 members. The 6 members are our vertices. We give them membership values according to their individual performance. There edge values are defined as their work performance with other member as a group. The positive membership value of incidence pair value defines the working efficiency of two employees in a company and negative membership value defines their loss possibility in the working due to controversial issues among the employees as a group. By the matching process, we will get the best match of partners.
Let be the BFIG and our destination is to obtain the matching with MMBFIN of
.
Step—1 Let the matching be empty by default for a BFIG
.
Step—2 The incidence pair (v1, v1v2) is elected incidence pair by using the criteria of selection of the vertices as mentioned above.
Step—3 The matching is obtained.
Step—4 By using this step, we obtained and
.
Step—5 Lastly, the M1 is selected as shown in Fig 13 and MMBFIN is computed as .
By using this process, the best partners are selected which gives us maximum working efficiency and the chances of loss due to any controversial issues among the employees in a company are also minimized.
5 Comparative analysis
In Fig 7, there were three jobs {u1, u2, u3} and four applicants {v0, v1, v2, v3}. Our target is to give maximum jobs to the applicants in order to get maximum working efficiency and minimizing the low performance due to controversial issues among the applicants. By using the matching concept, we obtained MMVBFIN = (2.7, −2.1) and got a matching for BBFIG by using vertices. Now, by using Fig 12 we obtained MMBFIN = (1, −0.6) and got the same matching
for BBFIG by using the incidence pairs. So, the result is better by using the incidence pairs as by using the vertices we have more chances of controversial issues i.e.,
and by using the incidence pairs, we have
.
Now, there are six employees in a company. We give them membership values according to their individual performance. There edge values are defined as their work performance with other member as a group. The positive membership value of incidence pair value defines the working efficiency of two employees in a company and negative membership value defines their loss possibility in the working due to controversial issues among the employees as a group. In Fig 11 we obtained MMVBFIN = (4.2, −1.8) and got a matching for BFIG by using vertices. Now, by using Fig 13 we obtained MMBFIN = (2, −1.2) and got the same matching
for BFIG by using the incidence pairs.
Both matchings either by using vertex or incidence pairs are same but incidence pairs are representing the influence of on vertex to other. So, the incidence graphs are more better as by using the incidence pairs we can see that which employee have greater influence on other or which group have better efficiency level.
6 Conclusion
Graph theory is very needful for presenting the data of real life problems. In this article, we enhanced the theory of BFIGs. The matching concept becomes very useful when it is discussed by using BFIGs because it also includes the controversial issues or chances of loss among the employees in a company. After introducing the concept of matching in BFIGs, its related propositions, results and theorems with some examples are presented. Matching numbers are obtained to improve the working quality of the employees in a company. Finally, a decision making graph of a company is presented to reflect the working of the members and achieving maximum results by minimizing chances of loss. Our goal is to enhance this research to soft FIGs, q-rung FIGs with more theorems and applications in forthcoming articles.
Acknowledgments
The authors are highly grateful to Editor-in-Chief and the anonymous reviewers for their valuable comments and suggestions to improve the quality of our manuscript.
References
- 1. Zadeh L. A., Fuzzy sets, Inf. Comput. 1965, 8, 338–353.
- 2.
Rosenfeld A.; Zadeh L. A; Fu K. S and Shimura M., Fuzzy graphs, Fuzzy Sets and their Applications, Academic Press, New York, 1975, 77–95.
- 3.
Mordeson J. N and Nair P. S, Fuzzy graphs and Fuzzy Hyper graphs, Physica-Verlag, Heidelberg, 1998, second edition, 2001.
- 4. Dinesh T., Fuzzy incidence graph-an introduction. Adv. Fuzzy Sets Syst. 2016, 21, 33–48.
- 5. Mathew S. and Mordeson J. N, Connectivity concepts in fuzzy incidence graphs. Inf. Sci. 2017, 382–383, 326-333.
- 6. Fang J.; Nazeer I.; Rashid T. and Liu J. B, Connectivity and Wiener index of fuzzy incidence graphs. Mathematical Problems in Engineering, 2021, 1–7.
- 7.
Zhang W. R, Bipolar fuzzy sets, Fuzzy Systems Proceedings, IEEE World Congress on Computational Intelligence, IEEE International Conference, 1998, 1, 835-840.
- 8. Akram M., Bipolar fuzzy graphs, Inf. Sci. 2011, 181, 5548–5564.
- 9. Akram M., Bipolar fuzzy graphs with applications, Knowledge- Based Syst. 2013, 39, 1–8.
- 10. Akram M. and Karunambigai M. G, Metric in Bipolar Fuzzy Graphs, World Appl. Sci. J. 2011, 14, 1920–1927.
- 11. Akram M., Dudek W. A, Regular bipolar fuzzy graphs, Neural Comput. Appl. 2012, 21, 197–205.
- 12. Akram M.; Li S. -G. and Shum K. P, Antipodal bipolar fuzzy graphs, Italian Journal of, Pure Appl. Math. 2013, 31, 97–110.
- 13. Akram M.; Dudek W. A and Sarwar S., Properties of bipolar fuzzy hyper graphs, Italian Journal of, Pure Appl. Math. 2013, 31, 426–458.
- 14.
Akram M., Sarwar M. and Dudek W. A, Graphs for the analysis of bipolar fuzzy information, Springer, 2021.
- 15. Mohanta K., Dey A., Pal A., Long H. V and Son L. H, A study of m-polar neutrosophic graph with applications. Journal of Intelligent and Fuzzy Systems, 2020, 38, 4, 4809–4828.
- 16. Xiao W., Dey A. and Son L. H, A study on regular picture fuzzy graph with applications in communication networks, Journal of Intelligent and Fuzzy Systems, 2020, 39, 3, 3633–3645.
- 17. Gong S. and Hua G., Topological indices of bipolar fuzzy incidence graph, Open Chemistry, 2021, 19, 1, 894–903.
- 18. Shen C. C and Tsai W. H, A graph matching approach to optimal task assignment in distributed computing systems using a minimax criterion. IEEE Trans. Comput, 1958, 34, 3, 197–203.
- 19.
Ramakrishnan P. V and Vaidyanathan M., Matching in fuzzy graphs, In Proceedings of National Conference on Discrete Mathematics and its Applications, 2007.
- 20. Mohan R. and Gupta A., Graph matching algorithm for task assignment problem. Int. J. Comput. Sci. Eng. Appl, 2011, 1, 6, 105–118.
- 21. Khalili M.; Borzooei R. A and Deldar M., Matching numbers in fuzzy graphs. Journal of Applied Mathematics and Computing. 2021, 67, 1, 1–22.