Abstract
The correspondence between a vertex and its neighbors has an essential role in the structure of a graph. Type-2 soft sets are also based on the correspondence of primary parameters and underlying parameters. In this study, we present an application of type-2 soft sets in graph theory. We introduce vertex-neighbors based type-2 soft sets over (set of all vertices of a graph) and (set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. Finally, we describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.
1. Preliminaries and Introduction
A graph consists of a nonempty set of objects , called vertices, and a set of pair of the element of , called edges. Two vertices and are adjacent if . A simple graph is an unweighted, undirected graph containing no multiple edges or graph loops. A graph is said to be a subgraph of if and The neighborhoods of a vertex in a graph are the set of all the vertices adjacent to including itself. The neighbors of a vertex in a graph are the set of all the vertices adjacent to excluding itself. The eccentricity of the vertex is the maximum distance from to any vertex. The radius of a graph is the minimum eccentricity of any vertex
Graph theory is rapidly moving into the mainstream of mathematics, mainly because of its applications in diverse fields which include electrical engineering (communications networks and coding theory), computer science (algorithms and computations), and operations research (scheduling). In many cases, it is important to deal with uncertainties using fuzzy sets and logics. The fuzzy graph is a weighted graph which gives a normalized relational strength over a fuzzy subset of a set [1]. Kauffman [2] gave the fundamental definition of a fuzzy graph on the basis of Zadeh’s fuzzy relations [3]. Rosenfeld [1] introduced another elaborated definition of fuzzy graphs. Subsequently, Bhattacharya [4] gave some useful results on fuzzy graphs and Mordeson and Nair [5] investigated some operations on fuzzy graph theory. In recent decades, fuzzy graphs were studied by many researchers. Akram et al. extended the concept of fuzzy graphs to bipolar fuzzy graphs [6], intuitionistic fuzzy hypergraphs [7], intuitionistic fuzzy graphs structures [8], and interval valued fuzzy graph [9]. Samanta et al. [10–12] studied vague graphs, fuzzy planar graphs, completeness and regularity of generalized fuzzy graph, and irregular bipolar fuzzy graphs.
Soft set theory [13], firstly initiated by Molodtsov, is a new mathematical tool for dealing with uncertainties. Indeed, Molodtsov’s approach shows the applicability of soft sets to several fields and provides some fundamental results subsequently augmented by works like Maji et al. [14] and Aktaş and Çağman [15], among others. Ali et al. [16] presented some new operations in soft set theory and based on the analysis of several operations on soft sets Sezgin and Atagün [17] studied the theoretical aspect of the soft set theory.
Let be a set of parameters that can have an arbitrary nature (numbers, functions, sets of words, etc.). Let be a universal set and the power set of is denoted by The formal definition of soft is defined as follows.
Definition 1 (see [14]). A pair is called a soft set over , where is a mapping given by
We refer to Molodtsov’s soft sets as type-1 soft sets (briefly T1SS). Note that the set of all T1SS over will be denoted by We refer to [13, 16, 18] for basic notions of soft set theory.
The applications of the soft set are progressing rapidly and many researchers are focusing on real and practical problems. Interestingly, some progress of soft sets has developed with graph theory. Ali et al. [19] introduced a new representation of graphs based on neighborhoods and soft sets. Akram and Nawaz presented the concepts of soft graphs [20, 21] and fuzzy soft graphs [22]. Let be a simple graph; let be any nonempty set. Let be an arbitrary relation from to A mapping from to is denoted as and defined as and a mapping from to is denoted as and defined as . Then is a T1SS over and is a T1SS over . The notion of a soft graph is defined as follows.
Definition 2 (see [21]). A -tuple is called a soft graph if it satisfies the following conditions:(i) is a simple graph.(ii) is a nonempty set of parameters.(iii) is a T1SS over .(iv) is a T1SS over .(v), for all , represents a subgraph of .The soft graph can also be written as .
In the rest of the paper, soft graph will be written as type-1 soft graph (briefly, T1SG). Some operations of T1SG are defined as follows.
Definition 3 (see [20]). Let and be two T1SG of Then is a type-1 soft subgraph of if(i),(ii)for each , is a subgraph of .
Definition 4 (see [20]). Let and be two T1SG of The extended union of and , denoted by , where , is defined, , asIt can be written as
Definition 5 (see [20]). Let and be two T1SG in The restricted intersection of and , denoted by , where , is defined, , as ,
It can be written as
Definition 6 (see [20]). Let and be two T1SG in The operation of and , denoted by , where , is defined, , as ,
Definition 7 (see [20]). Let and be two T1SG in . The operation of and , denoted by , where , is defined, , as ,
Definition 8 (see [20]). Let and be two T1SG of and , respectively, such that The Cartesian product of and is denoted and defined by , , , where and are the subgroups of and , respectively.
Note that denotes the Cartesian product of two crisp subgraphs. That is, .
For basic definitions of graph and soft graph, see [20, 21, 23–25].
In order to deal with associations between parameters, Chatterjeea et al. [26] propose the concept of type-2 soft set which is a generalization of Molodtsov’s soft set. It involves parameterization over an already parameterized set and hence has more freedom and efficiency compared to usual soft sets (termed as type-1 soft sets) in handling impreciseness. In fact, type-2 soft sets are a new approach to managing uncertainty. Interestingly, Chatterjeea et al. [26] investigate some basic operations on type-2 soft sets and use the model of type-2 soft sets in decision making problems. Recently, distance, entropy, and similarity measures of type-2 soft sets are introduced by Chatterjeea et al. [27].
Let be universe set and let be the set of parameters.
Definition 9 (see [26]). Let be a soft universe and let be the collection of all T1SS over . Then a mapping , is called a type-2 soft set (briefly T2SS) over and it is denoted by . In this case, corresponding to each parameter , is a T1SS. Thus, for each , there exists a T1SS, such that , where and . In this case, we refer to the parameter set as the “primary set of parameters,” while the set of parameters is known as the “underlying set of parameters.”
In this study, we present an application of type-2 soft set in graph theory. We introduce vertex-neighbors based type-2 soft sets over (set of all vertices of a graph) and (set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. We describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.
2. Type-2 Soft Graphs
The behavior and the selections of initial parameters and associated parameterized sets corresponding to initial parameters make the study of type-2 soft sets very interesting. Motivated by this, we study type-2 soft sets in graph theory and related properties. In a graph, every vertex regarding its location consists of a special relationship with its neighbors. Suppose that a company makes an analysis on the flow of transportation in two cities “” and “” of a country. We consider edge (line segment (link) between locations, that, is road) and vertex (location on the transportation network of interest (node), that is, town and road intersections). The company must make a survey on the flow of transportation from neighbor cities to control transportation in “” and “” and obtain a graphical representation to analyze the flow of transportation. The vertex-neighbors correspondence has an essential role in the structure of a graph. The type-2 soft set is also based on the correspondence of initial parameters and underlying parameters. The set of all neighbors of an element is defined as follows.
Definition 10. Let be a simple graph. The set of neighbors of an element is denoted by and defined by
Let be a simple graph and . Then Let a subset of be an arbitrary relation from to . The set of all T1SS over is denoted by and the set of all T1SS over is denoted by
Definition 11. Let be a simple graph; let and be the collection of all T1SS over . Then a mapping is called a T2SS over and it is denoted by . In this case, corresponding to each vertex , is a T1SS such that , where can be defined as , for all , and is the set of all neighbors of . This T2SS is also called a vertex-neighbors induced type-2 soft set (briefly, VN-type-2 soft set) over .
Definition 12. Let be a simple graph; let and be the collection of all T1SS over . Let be a VN-type-2 soft set over . Then a mapping is called a T2SS over and it is denoted by . In this case, corresponding to each vertex , is a T1SS such that , where can be defined as , for all , and is the set of all neighbors of . This T2SS is also called a VN-type-2 soft set over .
As it is mentioned in the above definitions, and are T1SS over and over , respectively. If represents a subgraph of , then by the definition of T1SG will be a T1SG of . This T1SG can also be represented by
2.1. Type-2 Soft Graph
A 5-tuple is called a type-2 soft graph (briefly, T2SG) if it satisfies the following conditions:(i) is a simple graph.(ii) is a nonempty set of parameters.(iii) is a VN-type-2 soft set over .(iv) is a VN-type-2 soft set over .(v)T1SS corresponding to , represents a type-1 soft graph.
A T2SG can also be represented by , where such that , The set of all T2SG of is denoted by .
Although the above representation of graphs provides a new framework for the applications of graph theory, it is restricted to the information from neighbors. In this regard, a likeness between parameterized vertices appears in some cases such as distinction between T2SG and is limited toward respective T1SG. However, T2SG is more general and hence a more efficient approach in graph theory as compared to T1SG.
Definition 13. A T2SG is said to be vertex-induced if for all .
Definition 14. A T2SG is said to be edge-induced if for all .
Example 15. Consider a graph as shown in Figure 1.
Let . Then , . Let and be two T2SS over and , respectively, such that and for all . Let , and , . ThenFigure 2 shows the respective T1SG corresponding to and , respectively. Hence, is a T2SG of . It is also called VN-type-2 soft graph.
Example 16. Consider a crisp graph such that and . Let be a nonempty set of parameters and , , and . We may define the pair of T2SS as follows:
Then the subgraphs of corresponding to parameters (vertices) , , and are shown in Figure 3. Hence, is a T2SG of . Consider . We may symbolize , as and denote a set of associations of , as . Then tabular representation of T2SG is given in Table 1.
3. Operations on Type-2 Soft Graphs
In this section, we present type-2 soft subgraph of a T2SG, union, intersection, OR operation, AND operation, and Cartesian product of T2SG.
Definition 17. Let and be two T2SG of Then is a type-2 soft subgraph of if(i),(ii)for each , T1SG corresponding to is a type-1 soft subgraph of T1SG corresponding to .
Example 18. Consider a graph as shown in Figure 4.
Let and . Then , , and . Let and be two T2SS over and , respectively, such that and for all , where and for all . Now, T2SS and are as in the following:Thus, is a T2SG as shown in Figure 5.
Let and be two T2SS over and , respectively, such that and for all , where , and for all . Then T2SS and are as in the following:Therefore, is a T2SG of as shown in Figure 6.
One can easily check that is a type-2 soft subgraph of
Theorem 19. Let and be two T2SG of Then is a type-2 soft subgraph of if and only if and for all .
Proof. Suppose is a T2SG of . Then by the definition of T2SG, (i),(ii)for each , T1SG corresponding to is a type-1 soft subgraph of T1SG corresponding to .Since T1SG corresponding to is a type-1 soft subgraph of T1SG corresponding to for all . Then and for all .
Conversely, and for all . As is a T2SG of , T1SS corresponding to is a T1SG of for all Also, is a T2SG of ; T1SS corresponding to is a T1SG of for all This implies that T1SG corresponding to is a type-1 soft subgraph of T1SG corresponding to for all . Thus, is a type-2 soft subgraph of .
Definition 20. Let and be two T2SG of The union of and , denoted by , where , is defined, , aswhere for all refers to the usual type-1 soft union between the respective T1SS corresponding to and , respectively. And,where for all refers to the usual type-1 soft union between the respective T1SS corresponding to and , respectively.
It can be written as
Example 21. Consider a graph as shown in Figure 7
Let and . Then , , and . Let and be two T2SS over and , respectively, such that and for all , where and for all . Now, T2SS and are as in the following:Thus, is a T2SG of .
Let and be two T2SS over and , respectively, such that and for all , where , and for all . Then T2SS and are as in the following:Therefore, is a T2SG of
Then union of and is , where andThe union of and is shown in Figure 8.
Note that, in the definition of T2SG, subgraphs in T1SG corresponding to may be connected or disconnected subgraphs of a simple graph. However, in this section we are now focusing only on connected subgraphs of a simple graph.
Theorem 22. Let and be two T2SG of with . Let T1SS corresponding to and be and , respectively. If for each , , then the union of and is a T2SG of
Proof. The union of and is defined as , where for all ,where for all refers to the usual type-1 soft union between the respective T1SS corresponding to and , respectively. And,where for all refers to the usual type-1 soft extended union between the respective T1SS corresponding to and , respectively.
is connected T2SG of . Then T1SS corresponding to is a connected T1SG of for all .
is connected T2SG of . Then T1SS corresponding to is a connected T1SG of for all .
Let , . Respective T1SS corresponding to and are connected T1SG of . Given that , this implies that T1SS corresponding to is connected T1SG of . As was taken to be an arbitrary element, theorem holds for all . Thus, is a T2SG of .
Lemma 23. Let and be two T2SG of . If , then their union is a T2SG of .
Definition 24. Let and be two T2SG of The intersection of and , denoted by , where , is defined by and for all , where (i) for all refers to the usual type-1 soft intersection between the respective T1SS corresponding to and , respectively,(ii) for all refers to the usual type-1 soft intersection between the respective T1SS corresponding to and , respectively.It can be written as
Example 25. Consider T2SG and defined in Example 18. The intersection between and is equal to as shown in Figure 6.
Definition 26. Let and be two T2SG of The operation of and , denoted by defined by , for all , where for all , refers to the usual type-1 soft operation between the respective T1SS corresponding to and , respectively, and for all refers to the usual type-1 soft operation between the respective T1SS corresponding to and , respectively.
Example 27. Consider graph that is shown in Figure 7. Let and . Then , , and . Let and be two T2SS over and , respectively, such that and for all , where and for all . Now, T2SS and are as in the following:Then is T2SG of .
Let and be two T2SS over and , respectively, such that and for all , where and for all . Then T2SS and are as in the following:Then is T2SG of .
The operation of and is , where andThe operation of and is shown in Figure 9.
Theorem 28. Let and be two T2SG of . Let T1SS corresponding to and be and for all , respectively, and let T1SS corresponding to and be and for all , respectively. For each , and . Then is a T2SG of
Proof. The operation of and is denoted by and defined by and for all , where for all refers to the usual type-1 soft operation between the respective T1SS corresponding to and , respectively, and for all refers to the usual type-1 soft operation between the respective T1SS corresponding to and , respectively.
is connected T2SG of . Then T1SS corresponding to is a connected T1SG of for all .
is connected T2SG of . Then T1SS corresponding to is a connected T1SG of for all .
Let . It may be mentioned here that , , such that . Respective T1SS corresponding to and are connected T1SG of . Given that and , this implies that is a T1SG of . Hence, T1SS corresponding to is connected T1SG of . As was taken to be an arbitrary element, theorem holds for all
Thus, is a T2SG of .
Definition 29. Let and be two T2SG of The operation of and is denoted by and defined by and for all , where for all refers to the usual type-1 soft operation between the respective T1SS corresponding to and , respectively, and for all refers to the usual type-1 soft operation between the respective T1SS corresponding to and , respectively.
Example 30. Consider graph as shown in Figure 10.
Let and . Then , , and . Let and be two T2SS, respectively, over and such that and for all , where and for all . Now, T2SS and are as in the following:Then is T2SG of
Let and be two , respectively over and such that and for all , where and for all . Then T2SS and are as in the following:Then is T2SG of
The operation of and is , where andThe operation of and is shown in Figure 11.
Theorem 31. Let and be two T2SG of . Let T1SS corresponding to and are and for all , respectively. For each , . Then is a T2SG of
Proof. Omitted
Definition 32. Let and be two T2SG over and , respectively, such that and . The Cartesian product of and is denoted by and defined as , where for all refers to the usual type-1 soft graph Cartesian product between the respective T1SS corresponding to and , respectively.
Example 33. Consider two graphs and as shown in Figures 12 and 13, respectively. Let and . Then , , and . Let and be two T2SS, respectively, over and such that and for all , where and for all . Now, T2SS and are as in the following:Then is T2SG of
Let and be two T2SS, respectively, over and such that and for all , where and for all . Then T2SS and are as in the following:Then is T2SG of
The Cartesian product of and is , where and is shown in Figure 14. Similarly, we can compute and show the respective graphs.
Theorem 34. Let be the Cartesian product of two simple graphs and . Let and be two T2SG of and , respectively. Then is a T2SG of .
Proof. The Cartesian product of two T2SG and is , where for all refers to the usual type-1 soft graphs Cartesian product between the respective T1SS corresponding to and , respectively.
Given that is a T2SG of , then T1SG corresponding to is a connected T1SG of . Also, given that is a T2SG of , then T1SG corresponding to is a connected T1SG of . This implies that T1SG corresponding to is a Cartesian product of two connected T1SG. Therefore, T1SG corresponding to is connected T1SG of , for all . Thus, is a T2SG of .
4. An Application of Type-2 Soft Graphs
It is well known that graph theory has many applications in computer networks and communication networks. In this manner, our approach is noteworthy to enhance the study of networks. In this section, we contemplated an application of T2SG in communication networks.
It is necessary to explain some new notions of T2SS that will be used in this section. Let and be two T2SS over . Let, for each , and be the T1SS corresponding to and , respectively. A correlation table of T2SS is an table in which the entry is a resultant of for all ( and ). There are , partitions in correlation table, where each partition represents an operation on respective T1SS corresponding to and for all . The -union in a partition is the row-wise union of entries and the -intersection in a partition is the column-wise intersection of entries.
Consider a communication network of 14 communications towers (see Figure 15), where each tower is represented as a vertex and each communication link between towers is represented with an edge in a resulting graph. The communication among the towers in the network is done by using two types of messages: Hello (used for sending information to neighbors) and Data (used for data transmission). There are two types of towers in the network: (i) working as a base station and working in subnetworks and (ii) working in subnetworks. Hello, messages are only considered by one-hop neighbors, while the Data messages are representing sending and gathering information from subnetworks and therefore must be transmitted in a multihop manner towards the base stations and towards the subnetworks.
Two communications companies are using this network of towers, and each one uses a set of base stations for communication: the first company is using towers , as base stations, while the second company is using towers , as base stations. Let and . For any base station messaging is as follows:(i) sends Hello messages and their neighbors send these messages as Data messages towards the set of all towers in the network.(ii)Also, the neighbors of gather messages from the whole network and then send these messages as Hello messages to .To analyze the weak service (messaging), both companies make a survey on their sets of base stations. The region (towers and the links between towers) of weak service of and may be defined in the form of T2SG.
The region of weak service for is as follows: for base station , region of weak messaging is defined as a T1SS, and .
For base station , region of weak messaging is defined as a T1SS, and Then T2SS are as follows:One can check that is a T2SG of .
The region (towers and the links between towers) of weak service for is as follows: for base station , region of weak messaging is defined as a T1SS, and
For base station , region of weak messaging is defined as a T1SS, and Then T2SS are as follows:
One can check that is a T2SG of .
To analyze the region (towers and the links between towers) in which both the sets of base stations are incorporated for very fragile service in the communication network, first, construct correlation tables of and for vertices and edges, respectively. Let . In the correlation Table 2 entry is a resultant of and in the correlation Table 3 entry is a resultant of for all ( and ). There are , partitions in the correlation Table 2, where each partition represents an operation on respective T1SS corresponding to and , respectively. Also, there are , partitions in the correlation Table 3, where each partition represents an operation on the respective T1SS corresponding to and , respectively.
Take -union of each partition in Table 2 and represent it in Table 4. Similarly, compute Table 5 from Table 3.
Next, take -intersection on each partition in Table 4 and represent it in Table 6. Similarly, compute Table 7 from Table 5.
Take -union on Table 6 and represent it in matrix of vertices. Sequentially, obtain a matrix of edges from Table 7.
Select the subgraph that is involved in extremely poor communications for both companies integratedly. One can easily check that the vertices , , , and appear maximum times (twice) in matrix of vertices and the edges and appear twice in matrix of edges. Thus, the very fragile communication for both companies integratedly is shown in Figure 16. Eventually, both companies are incorporated to fix the fragile communication on resultant subgraph.
We now present the general procedure of our method which is used in our application from the following algorithm.
Algorithm(i)Input the two sets of base stations in a communication network.(ii)Define T2SG on each set of base stations, according to the weak communication messaging, that is, Hello messages and Data messages.(iii)Take operation on T2SG represents it in the correlation tables:(1)Table 1 represents operation on two T2SS of the vertices and it contains partitions, where each partition is an operation of respective TISS corresponding to and , respectively, (2)Table 2 represents operation on two T2SS of the edges and it contains partitions, where each partition is an operation of respective TISS corresponding to and , respectively, (iv)Take -union on each partition of Table 1, representing it in Table 3. Similarly, compute Table 4 from Table 2.(v)Take -intersection on each partition in Table 3, representing it in Table 5. Similarly, compute Table 6 from Table 4.(vi)Take -union on Table 5 represents an matrix of vertices. Similarly, obtain an matrix of edges from Table 6.(vii)Find the vertices that appear maximum “” times in the matrix of vertices, and then find the edges that appear “” times in the matrix of edges. Note that .(viii)By using vertices and edges that are obtained from step (vii), represent a graph of very unwell communications for both sets of base stations integratedly. It may be a connected or disconnected graph.Finally, both companies incorporated to fix the fragile communication on resultant subgraph. Similarly, we can present an algorithm for more than two sets of base stations in a network.
5. Conclusion
In the above study, we have presented an application of type-2 soft sets in graph theory. We have introduced vertex-neighbors based type-2 soft sets over and . We have introduced some operations on type-2 soft graphs. We have described a practical application of type-2 soft graphs in communication networks and present procedure as an algorithm. The type-2 soft graphs are more general and hence a more efficient tool as compared to type-1 soft graphs. The idea presented in this paper has practical applications in the social networks and computer networks.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Guangdong Province (2016A030313552 and 2016A030307037), the Guangdong Provincial Government to Guangdong International Student Scholarship (yuejiao 187), and Guangzhou Vocational College of Science and Technology (nos. 2016TD03 and 16YJAZH081).