Abstract
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. Recently, type 2 soft graphs have been introduced. Structurally, it is a very efficient model of uncertainty to deal with graph neighbors and applicable in applied intelligence, computational analysis, and decision-making. The present paper characterizes type 2 soft graphs on underlying subgraphs (regular subgraphs, irregular subgraphs, cycles, and trees) of a simple graph. We present regular type 2 soft graphs, irregular type 2 soft graphs, and type 2 soft trees. Moreover, we introduce type 2 soft cycles, type 2 soft cut-nodes, and type 2 soft bridges. Finally, we present some operations on type 2 soft trees by presenting several examples to demonstrate these new concepts.
1. Preliminaries and Introduction
A graph consists of a nonempty set of objects , called vertices, and a set of two element subsets of called edges. Two vertices and are adjacent if . A loop is an edge that connects a vertex to itself. 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 graph neighborhoods of a vertex in a graph is the set of all the vertices adjacent to including itself. The graph 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 distance between two vertices and in a graph is the number of edges in a shortest path, denoted by . The radius of a graph is the minimum eccentricity of any vertex A graph is called a tree if it is connected and contains no cycles. Equivalently (and sometimes more useful), a tree is a connected graph on vertices with exactly edges. A forest is a disjoint union of trees. The degree of a vertex of a simple graph is the number of edges incident to the vertex. A regular graph is a graph where each vertex has the same number of neighbors. A graph that is not a regular graph is called irregular graph. A graph is called neighborly irregular graph if no two adjacent vertices have the same degree. A complete graph is a graph in which each pair of graph vertices is connected by an edge. For basic definitions of graphs see [1–3].
Soft set theory [4], firstly initiated by Molodtsov, is a new mathematical tool for dealing with uncertainties. Some fruitful operations, soft set theory, are presented by Maji et al. [5] and Ali et al. [6]. We refer to Molodtsov’s soft sets as type 1 soft sets briefly Let be a set of parameters that can have an arbitrary nature (numbers, functions, sets of words, etc.). Let be a universe and the power set of is denoted by The soft set is defined as follows.
Definition 1 (see [5]). A pair is called a soft set over , where is a mapping given by
Note that the set of all over will be denoted by Many researchers take attention at applicability of soft sets in real and practical problems. In recent years, research on soft set theory has been rapidly developed, and great progress has been achieved, including works of soft sets in graph theory. Ali et al. [7] introduced a representation of graphs based on neighborhoods. Akram et al. introduced the concepts of soft graphs [8, 9] and soft trees [10]. Let be a simple graph, be any nonempty set.
Let be an arbitrary relation from to A mapping from to denoted as and defined as and a mapping from to denoted as and defined as Then is a over and is a over The notion of a soft graph is defined as follows.
Definition 2 (see [9]). A -tuple is called a soft graph if it satisfies;(i) is a simple graph.(ii) is a non-empty set of parameters.(iii) is a over .(iv) is a over .(v) for all , represents a subgraph of .
The soft graph can also be written as , where
Definition 3 (see [9, 10]). Let be a soft graph of . Then is said to be a soft tree (resp., regular soft graph, irregular soft graph, neighborly irregular soft graph, soft cycle) if every is a tree (resp., regular graph, irregular graph, neighborly irregular graph, and cycle) for all
A soft graph is called a regular soft graph of degree if is a regular graph of degree for all . In the rest of paper soft tree and soft cycle will be written as type 1 soft tree (briefly, ) and type 1 soft cycle (briefly, ), respectively. Some operations of are defined as follows.
Definition 4 (see [10]). Let and be two of Then is a type 1 soft subtree of if(i);(ii)for each , is a subtree of .
Definition 5 (see [10]). Let and be two of The extended union of and , denoted by , where , is defined , asIt can be written as
Definition 6 (see [10]). Let and be two in The restricted intersection of and , denoted by , where , is defined , as ,
It can be written as
Definition 7 (see [10]). Let and be two in The operation of and , denoted by , where , is defined , as ,
Definition 8 (see [10]). Let and be two in The operation of and , denoted by , where , is defined , as ,
Definition 9 (see [8]). Let be a soft graph of . Then the complement of is denoted by and defined by , where and contain those edges which are not included in .
A generalization of soft sets called type 2 soft sets is introduced by Chatterjee et al. [11]. Some results on type 2 soft sets are validated by Yang and Wang [12] and some new operation on typ-2 soft sets are defined by Khizar et al. [13]. Let be universe set and be the set of parameters. The definition of type 2 soft set is as follows:
Definition 10 (see [11]). Let be a soft universe and be the collection of all over . Then a mapping , is called a type 2 soft set (briefly ) over and it is denoted by . In this case, corresponding to each parameter , is a . Thus, for each , there exists a , 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”.
Let be a simple graph. The set of all over is denoted by and the set of all over is denoted by The set of neighbors of an element is denoted by and defined by Then , where . Let a subset of be an arbitrary relation from to . Let be a simple graph. Khizar et al. [14] introduced type 2 soft graphs by using over and over .
Definition 11 (see [14]). Let be a simple graph, and be the collection of all over . Then a mapping is called a over and it is denoted by . In this case, corresponding to each vertex , is a . Thus, for each , there exists a , such that where can be defined as , for all and is the set of all neighbors of . This is also called a vertex-neighbors induced type 2 soft set (briefly, -type 2 soft set) over
Definition 12 (see [14]). Let be a simple graph, and be the collection of all over . Let be a -type 2 soft set over . Then a mapping is called a over and it is denoted by . In this case, corresponding to each vertex , is a . Thus, for each , there exists a , such that where can be define as , for all and is the set of all neighbors of . This is also called a -type 2 soft set over
Definition 13 (see [14]). A -tuple is called a type 2 soft graph (briefly, ) if it satisfies following conditions:(i) is a simple graph.(ii) is a non-empty set of parameters.(iii) is a -type 2 soft set over .(iv) is a -type 2 soft set over .(v) corresponding to , represents a type 1 soft graph. A can also be represented by , where such that ,
2. Certain Types of Type 2 Soft Graphs
In this section, we present regular type 2 soft graphs, irregular type 2 soft graphs, and type 2 soft trees. Moreover, we introduce type 2 soft cycles, type 2 soft cut-nodes, and type 2 soft bridges.
Definition 14. Let be a of Then is said to be a regular if corresponding to every is a regular for all A is called a regular of degree if corresponding to is a regular of degree for all
Example 15. Consider a graph as shown in Figure 1. Let . It may be written that and . Let and be two over and respectively, such that and for all . Define , for all , and , for all Then and are as in the following , , , , , . One can check that is a regular of as shown in Figure 2.
Theorem 16. Let be a complete graph. Then every of is a regular of .
Proof. Suppose is a complete graph and be a of . Let be a corresponding to for all Then is a subgraph of . Since every subgraph of a complete graph is complete and every complete graph is regular. Therefore, is a regular subgraph. This implies that corresponding to for all is regular . Hence is a regular of .
Note that if is a regular of then need not be a complete graph.
Example 17. In Example 15, is regular but is not a complete graph.
Definition 18. Let be a of . Then the complement of is denoted by and defined by for all , where corresponding to is the complement of corresponding to for all to .
Example 19. Consider a graph as shown in Figure 3. Let . It may be written that and . Let and be two over and respectively, such that and for all . Define , for all , and , for all Then and are as follows:Then and are as follows:The complement of is as shown in Figure 4.
Proposition 20. If is a regular of then is a regular of .
Proof. Let be a regular of . Let be a corresponding to for all Then is a regular subgraph of . Since complement of a regular graph is regular, is a regular subgraph. This implies that corresponding to for all is regular . Hence is a regular of .
Proposition 21. Let be a regular graph. Then every of may not be a regular of .
Example 22. Consider a regular graph , where and . Let . It may be written that and . Let and be two over and respectively, such that and for all . Define , for all , and , for all Then and are as follows:Then is not a regular of as shown in Figure 5.
Definition 23. Let be a of . Then is said to be irregular if corresponding to is an irregular for all .
Example 24. Consider a graph as shown in Figure 6. Let . It may be written that and . Let and be two over and respectively, such that and for all . Define , for all , and , for all Then and are as follows:Then is an irregular of as shown in Figure 7.
Definition 25. Let be a type 2 soft graph of . Then is said to be neighborly irregular if corresponding to is an neighborly irregular for all .
Example 26. Consider a graph as shown in Figure 8. Let . It may be written that and . Let and be two over and respectively, such that and for all . Define , for all , and , for all Then and are as in the following; , , , , Then is a neighborly irregular of as shown in Figure 9.
Definition 27. Let be a of a simple graph Let be a corresponding to for all An ede in is said to be a type 2 soft bridge if its removal disconnect the subgraph , .
Definition 28. Let be a of a simple graph Let be a corresponding to for all An vertex in is said to be a type 2 soft cut-vertex if its removal disconnect the subgraph , .
Example 29. Consider defined in Example 24. In the Figure 7, type 2 soft bridges of are in , in , in and in . Moreover, type 2 cut-vertices of are in , in , in and in .
Definition 30. Let be a of Then is said to be a type 2 soft tree (briefly, ) if corresponding to every is a for all
Example 31. Consider a graph as shown in Figure 10. Let and , . Let and be two neighbor-induced over and respectively, such that and for all . We define , for all and , for all Then and are as in the following , , , Therefore, is a of as shown in Figure 11. It is also called -type 2 soft tree.
Hence, is a of . It is also called -type 2 soft graph. It may be written that . We may symbolize , as and denote a set of associations of , as . Then tabular representation of is given in Table 1.
Theorem 32. Let be a corresponding to . Let be subgraph with vertices of and a of . Then is not a complete of .
Proof. Let be a corresponding to . Suppose on the contrary that is a complete , then every subgraph , for all is complete. Suppose be arbitrary nodes of and they are connected by an edge . Since is subgraph with vertices of , then we can always find at least one vertex which is connected to by an edge and to by an edge , because is a complete graph. Then there exists a cycle . Therefore, cannot be a which contradicts the fact that is a connected subgraph of . Hence, cannot be a complete .
Definition 33. Let be a and be a corresponding to for any . Then is called type 2 soft forest if consists of more than one disconnected tree for all .
Definition 34. Let be a of . Then is said to be a type 2 soft cycle (briefly, ) if corresponding to is a type 1 soft cycle of for each
Example 35. Consider a simple graph , where and , Let . It may be written that and . Let and be two over and , respectively, such that and for all . Define , for all , and , for all Then and are as follows:One can check that is a of as shown in Figure 12. It is also called -type 2 soft cycle.
Theorem 36. If is a of then is not a of .
Proof. Let be a of . Let be a type 1 soft cycle corresponding to . By definition, tree does not contain cycle. Then is not a tree for all , so that is not a type 1 soft tree. Hence is not a of .
The converse of above theorem is not true in general; i.e., if is not a then need not be a . The following example illustrates it.
Example 37. Consider a graph as shown in Figure 13.
Let . Then , . Let and be two over and , respectively, such that and for all . Let , and , . ThenFigure 14 shows the respective corresponding to and respectively. One can check that , , , and are not trees. This implies that is not a of . But is not a of .
Proposition 38. Every of is a regular of .
Proof. Suppose that is a . Let be a corresponding to for any . Then is a cycle for all . Since cycle is closed path and each vertex has degree , this implies that is a regular graph for all . Therefore is regular . Since was taken to be arbitrary, thus it holds for all . Hence is a regular of .
3. Operations on Type 2 Soft Trees
In this section, we present type 2 soft subtree of , union, intersection, operation, and operation of
Definition 39. Let and be two of Then is a type 2 soft subtree of if(i),(ii)for each , corresponding to is a type 1 soft subtree of corresponding to .
Example 40. Consider a simple graph , where and Let , . It may be written that , and .
Let and be two over and , respectively, such that and for all . Define , for all , , for all Then and are as follows:Then is a of as shown in Figure 15.
Let and be two over and , respectively, such that and for all . DefineThen and are as follows:Then is a of as shown in Figure 16.
One can check that and , . Hence is a type 2 subtree of .
Theorem 41. Let and be two of Then is a type 2 soft subtree of if and only if and for all .
Proof. Suppose is a type 2 soft subtree of . Then, by the definition of type 2 soft subtree,(i),(ii)For each , corresponding to is a type 1 soft subtree of corresponding to . Since corresponding to is a type 1 soft subtree of corresponding to for all . Then and for all .
Conversely, given that and for all . As is a of , corresponding to is a of for all Also, is a of , corresponding to is a of for all This implies that corresponding to is a type 1 soft subtree of corresponding to for all . Thus, is a type 2 soft subtree of .
Definition 42. Let and be two of The of and , is denoted by , where is defined , aswhere for all refers to the usual type 1 soft union between the respective corresponding to and , respectively. And,where for all refers to the usual type 1 soft extended union between the respective corresponding to and , respectively.
It can be written as
Theorem 43. Let and be two of with . Then is a of
Proof. The of and is defined as where for all ,where for all refers to the usual type 1 soft union between the respective corresponding to and , respectively. And,where for all refers to the usual type 1 soft extended union between the respective corresponding to and , respectively.
Since is a of . Then corresponding to is a of for all .
Since is a of . Then corresponding to is a of for all .
It is given that . Thus, is a of .
If then union of two may not be a as it can be seen in the following example.
Example 44. Consider a graph defined in Example 40. Let ,. It may be written that , and .
Let and be two over and , respectively, such that and for all . Define , for all and , for all Then and are as follows:Then is a of .
Let and be two over and , respectively, such that and for all . Define , for all and , for all . Then and are as follows:Then is a of . By the definition of of , By routine calculations, it is easy to see that corresponding to and are . But corresponding to is a disconnected type 2 soft forest, as shown in Figure 17. Therefore, is not a of .
Lemma 45. Let and be two of . If , then their union is a of .
Definition 46. Let and be two of The of and , denoted by , where is defined as , for all , where for all refers to the usual type 1 soft intersection between the respective corresponding to and , respectively. And for all where for all refers to the usual type 1 soft intersection between the respective corresponding to and , respectively.
It can be written as
The intersection of two may not be as it can be seen in the following example.
Example 47. Consider a simple graph shown in Figure 18. Let and . It may be written that , and .
Let and be two over and , respectively, such that and for all . Define , for all and , for all Then and are as follows:Then is a of .
Let and be two over and , respectively, such that and for all . Define , for all and , for all . Then and are as follows:Then is a of . By the definition of of , By routine calculations, it is easy to see that corresponding to is a disconnected , as shown in Figure 19. Therefore, is not a of .
Definition 48. Let and be two 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 corresponding to and respectively and for all refers to the usual type 1 soft operation between the respective corresponding to and respectively.
Example 49. Consider a simple graph , where and Let ,. It may be written that , and .
Let and be two over and , respectively, such that and for all . Define , for all and , for all Then and are as follows:Then is a of .
Let and be two over and , respectively, such that and for all . Define , for all . Then and are as follows:Then is a of . The operation on and is defined as in the following:The operation on and is shown in Figure 20.
Definition 50. Let and be two 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 corresponding to and respectively and for all refers to the usual type 1 soft operation between the respective corresponding to and respectively.
Example 51. Consider a simple graph defined in Example 49. Let , . It may be written that , and .
Let and be two over and respectively, such that and for all . Define , for all , , for all Then and are as follows:Then is a of .
Let and be two over and , respectively, such that and for all . Define , for all . Then and are as follows:Then is a of . The operation on and is defined as in the following:The operation on and is shown in Figure 21.
4. Conclusion
In above study, we have characterized type 2 soft graphs on underlying subgraphs (regular subgraphs, irregular subgraphs, cycles, trees) of a simple graph. We have presented regular type 2 soft graphs, irregular type 2 soft graphs, and type 2 soft trees. Moreover, we have introduced type 2 soft cycles, type 2 soft cut-nodes, and type 2 soft bridges. Finally, we have presented some operations on type 2 soft trees by presenting several examples to demonstrate these new concepts. In future work, we will extend our work in following areas of research:(i)Applications of type 2 soft graphs in computer networks and social networks(ii)Fuzzy type 2 soft graphs and their applications.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper is supported by the High Level Construction Fund of Guangzhou University China. Also this work was supported by the Natural Science Foundation of Guangdong Province (2016A030313552), the Guangdong Provincial Government to Guangdong International Student Scholarship (yuejiao 187), and Guangzhou Vocational College of Science and Technology (no. 2016TD03).