Abstract
This study investigates the domination, double domination, and regular domination in intuitionistic fuzzy hypergraph (IFHG), which has enormous application in computer science, networking, chemical, and biological engineering. Few properties of double domination and regular domination of IFHG are established. Furthermore, the definitions of complement and independent set of IFHG are given. The relation between the domination of an IFHG and the independent set of its complement was discussed. Moreover, the application of the double domination in the IFHG was illustrated by determining the containment zones for epidemic situations like COVID-19.
1. Introduction
Presently, the unavailability of complete information arises for complex processes in technology and science features. To handle such situations, in various types of uncertain elements of systems, the mathematical models need to be developed; a vast number of these models are based on an extension of the ordinary set theory to the fuzzy sets.
Zadeh [1] introduced the concept of a fuzzy set in 1965. Zhang et al. [2] analyzed the hesitant fuzzy preference relations. The theory and application of the Intuitionistic Fuzzy Sets were discussed by Atanassov [3]. In defining fuzzy sets, Atanassov added the new component degree of nonmembership to determine Intuitionistic Fuzzy Sets (IFS). The degree of membership and nonmembership is almost independent. However, the sum of these degrees should be less than or equal to one.
Graph theory has enormous applicability in the electrical industry, computer science, system analysis, economics, and biochemistry. Bondy and Murthy [4] discussed several concepts and applications of graph theory. The domination in graphs was examined by Cockayne et al. [5]. The uncertainty of graph-theoretic problems arises in several cases. In such cases, the uncertainty could be dealt with using fuzzy sets and Intuitionistic Fuzzy Sets. Nagoorgani and Sajith Begum [6] defined the degree, order, and size in Intuitionistic Fuzzy Graph (IFG) and extended the properties. New concepts in IFG were initiated by Shao et al. [7]. He elaborated on the application of the IFG in the water supply system. Hypergraphs are a generation of graphs in the case of a set of multiarray relations and have been considered a valuable tool for studying the system’s structure. The notion of hypergraph has extended with the fuzzy theory and Intuitionistic Fuzzy Sets as Intuitionistic Fuzzy Hypergraph (IFHG).
Moderson and Nair [8] defined fuzzy hypergraphs. Pradeepa and Vimala [9] examined the regular and totally regular intuitionistic fuzzy hypergraph (IFHG). Yahya and Mohammad Ali [10] described the max product of complement of IFG. Laqman et al. [11] studied the hypergraph representations of complex fuzzy information. Akram and Sarwar [12] discussed the applicability of m-polar fuzzy competition graphs. Akram and Nagoorgani [13] described the strong intuitionistic fuzzy graphs. Several researchers have contributed to the field of IFHG and elaborated its applications [14–17]. Domination in graphs has applicability in problems related to monitoring communication networks and application in LPG supply systems.
Nagoorgani et al. [18] described some exciting properties of the fuzzy dominating set, fuzzy independent set, and fuzzy minimal dominating set. He also established a new type of dominating fuzzy graphs. Domination in fuzzy graphs using strong edges was introduced by Nagoorgani and Chandrasekaran [19]. Somasundaram and Somasundaram [20] gave more concepts of independent domination and connected domination in the fuzzy graph. Nazeer et al. [21] discussed the domination of fuzzy incidence graphs with the algorithm and application for the selection of medical lab. Enriquez et al. [22] discussed the domination in fuzzy directed graphs. Finally, Rana [23] elaborated on a survey on the domination of fuzzy graphs.
Several researchers have examined the theory and application of domination graphs, fuzzy graphs, and IFHG [24–31]. However, no research has been established on domination, double domination, and regular domination in Intuitionistic Fuzzy Hypergraph (IFHG), which has enormous application in computer science, networking, chemical, and biological engineering. This study discussed domination, double domination, and regular domination in Intuitionistic Fuzzy Hypergraph (IFHG). Furthermore, the relation between the domination of an Intuitionistic Fuzzy Hyper Graph and the independent set of its complement was discussed and derived some results with proof and examples. Finally, the application for determining the containment zones for epidemic situations like COVID-19 is given.
2. Basic Definitions and Examples
This section gives some basic definitions of Intuitionistic Fuzzy Hypergraph (IFHG). Also, the definition of the double dominating fuzzy hypergraph, regular dominating, and regular double dominating set complement fuzzy hypergraph are introduced.
Definition 1. Consider the universal set . The fuzzy set in is represented by is the membership degree of element in the fuzzy set A.
Definition 2. For a fixed set be, an Intuitionistic Fuzzy set (IFS) A in is of the form , where the function and are determined as the “membership, and non-membership degree of element respectively” and for every ,
Definition 3. An Intuitionistic Fuzzy Graph (IFG) is of the form , where (i) such that and denote the degree of membership and nonmembership of the element , respectively, and 0 , for every (ii), where such that and are such thatand 0 for every , where .
Here, the degree of membership and degree of nonmembership of the vertex is denoted by the triple (, ). The degree of membership and degree of nonmembership of the edge relation on V is denoted by the triple .
Definition 4. The crisp subset of in which all its elements have nonzero membership degree is defined as the support of the fuzzy set :
Definition 5. An intuitionistic fuzzy hypergraph (IFHG) is an ordered pair , where (i) a finite set of vertices(iii) a finite intuitionistic fuzzy subset of V(iii) and (iv), (v)Here the edges are intuitionistic fuzzy sets. denote the degree of membership and degree of nonmembership of the vertex to the edge . Thus, the elements of the incidence matrix of intuitionistic fuzzy hypergraph are of the form (, ), the sets are crisp sets.
Definition 6. An intuitionistic fuzzy hypergraph is said to be an intuitionistic fuzzy subhypergraph (IFHSG) of , that is, and for every .
Definition 7. The order of an intuitionistic fuzzy hypergraph is the number of vertices of the number of hyperedges is called as the size of the intuitionistic fuzzy hypergraph and is denoted by .
Definition 8. The degree of a vertex in an Intuitionistic fuzzy Hyper graph is defined as the sum of the weights of the strong edges incident at . It is denoted by . The minimum degree of is . The maximum degree of is .”
Definition 9. An edge of an IFHG is semi–µ-strong IFHG if , where, .
Example 1. From Figure 1and Table 1, semi–µ-strong edges are , .
Definition 10. An edge of an IFHG is semi–γ-strong IFHG if .
Example 2. From Figure 2 and Table 2, semi–γ strong edges of are.
Definition 11. An edge of an IFHG is strong if , where , j = 1,2, ..., n.
Example 3. From Figure 3 and Table 3, the semi–µ-strong edges of are , and semi-γ-strong edges are , .
Therefore, the strong edges of are , , and .
Definition 12. Any two subsets of vertices of an IFHG are semi–µ-dominating each other if there exist a semi–µ-strong edge .
Definition 13. Any two subsets of vertices of an IFHG are semi–γ-dominating each other if there exist a semi γ-strong edge .
Definition 14. Any two subsets of vertices of an IFHG are dominating each other if there exist a strong edge .
Definition 15. Let be an IFHG, a subset of is called a dominating set of , if for every belongs to V-, there is u belongs to , such that and are strong neighbors.
Example 4. From Figure 4 and Table 4, the strong edges of the IFHG are , . The dominating sets of IFHG are .
Definition 16. An IFHG, is complete µ-strong IFHG, if and , where
Definition 17. An IFHG, is complete γ-strong IFHG, if and , where
Definition 18. An IFHG, is complete IFHG, if ) and , where
Definition 19. Let be an IFHG, a subset of is called independent set of if and , where .
Example 5. From Figure 5 and Table 5, the strong edges of the IFHG are , .
The independent sets of the IFHG are {....}
Definition 20. An independent set of an IFHG is said to be a maximal independent set, if for every vertex , the set is not an independent set.
Definition 21. Two vertices in an IFHG, are said to be independent set if there is no strong edge between them.
Definition 22. A vertex of an IFHG, are isolated vertices if and for all .(i.e.) .
Thus, the other vertices in should not be dominated by an isolated vertex.
Definition 23. The complement of an IFHG, is an IFHG , where (i)(ii) and for all (iii) and for all , .
Example 6. From Tables 6 and 7, it is found that IFHG in Figure 7 is the complement IFHG of Figure 6.
Definition 24. Let be an IFHG. A subset of is a double dominating set of , if for each vertex in V- is dominated at least by two vertices in is the minimum fuzzy cardinality of all the double dominating set of is defined as the double domination number of and denoted by .
Example 7. From Figure 8 and Table 8, the strong arcs are , , so the double dominating set of the IFHG is = , where V- =
Definition 25. A vertex is an end vertex of IFHG, and it has at most one strong neighbor in .
Definition 26. A vertex , is said to be a cut vertex in IFHG if deleting a vertex reduces the strength of the connectedness between some pair of vertices.
Definition 27. Let u be a vertex in an IFHG, the set is called neighborhood of
Definition 28. Let be an IFHG. A set subset of is called regular intuitionistic fuzzy hyper dominating set if (i)Every vertex in has strong edge to at least one vertex in .(ii)The hyperdegree of all the vertices in should be same.
Definition 29. Let be an IFHG. A set subset of is called regular double dominating set of intuitionistic fuzzy Hyper graph if (i)Every vertex in adjacent to at least two vertices in .(ii)All the vertices in has the same hyperdegree.
Example 8. From Figure 9 and Table 9, the strong arcs of the IFHG are ; hence, the regular double dominating set of is , here .
Definition 30. The minimum cardinality of regular intuitionistic fuzzy hyper dominating set is regular intuitionistic fuzzy hyper domination number, and it is denoted by .
Definition 31. Let be an IFHG. A set subset of is called minimal regular intuitionistic fuzzy hyper dominating set if (i)Any subset of is not a regular intuitionistic fuzzy hyper dominating set.(ii)All the vertices in has the same hyperdegree.
Definition 32. An independent set of an IFHG, is said to be regular independent intuitionistic fuzzy hyper set if a set is subset of , (i) and (ii)All the vertices in has hyperdegree and hyperedges.
Definition 33. Let be an IFHG. A set subset of is called maximal regular independent intuitionistic fuzzy hyperset if for every vertex the set is not a regular independent intuitionistic fuzzy hyperset.
3. Results and Proofs
Theorem 1. The double dominating set of an IFHG exists only if every vertex in contains at least two other vertices as strong neighbors.
Proof. Let is a double dominating set of an IFHG. If there exists a vertex in with a single strong neighbor, let it be and its strong neighbor is .
Case 1. If , then has no strong neighbor in , this implies that cannot be a double dominating set.
Case 2. If such that has exactly one strong neighbor, again, this implies that cannot be a double dominating set.
We obtain contradiction in both the cases. Hence, there exist at least two strong neighbors for every vertex in .
Example 9. From Figure 10 and Table 10, the strong arcs of IFHG are ; hence, , .
has at least two neighbors.
Theorem 2. The double dominating set of an IFHG is the set of all vertices of that IFGH if and only if all of its vertices are end vertices.
Proof. Let all the vertices of IFHG are end vertices. Since has only end vertices, all of its vertices have exactly one strong neighbor.
Also from Theorem 1, all the vertices in should have at least two strong neighbors, which imply that there are no vertices in double dominating set.
That is, , Hence, we have .
Example 10. From Figure 11 and Table 11, the strong arcs are , and all the vertices of are end vertices. Hence, the double dominating set of is itself.
Theorem 3. Let be an IFHG, and if the double dominating set of is an independent set of , then it was not an independent set of .
Proof. Let be a double dominating set of H. If is an independent set of . Let be a complement of intuitionistic fuzzy hypergraph . In , where , and for all , and , for all , . Here only the values of hyperedges are changed in , which implies most of the adjacent vertices in have strong neighbors and also different double dominating set exists in .
Therefore, the same double dominating set in cannot be an independent set of .
Theorem 4. A regular independent set is a regular maximal independent set of an IFHG if it is regular independent and dominating set of IFHG.
Proof. Let be a regular maximal independent set in an IFHG, then for every the set is not an independent set. It is trivial that is regular independent of . It is enough to prove that Regular dominating set, suppose that is not a regular dominating set, and then there exists at least one vertex , such that there is no strong neighbor for in , which implies that is an independent set of . This contradicts the fact that is a regular maximal independent set of an IFHG . Hence, is regular dominating and independent set of IFHG.
Theorem 5. A regular dominating set is a regular minimal dominating set of an IFHG if it is regular dominating and independent set of IFHG.
Proof. Let is regular minimal dominating set of . It is trivial that is regular dominating of It is enough to prove that Regular independent set, suppose that is not a regular independent set, then there exists at least one vertex such that at least one vertex in has a strong hyper-edge to V, which implies is regular dominating set. This contradicts the fact that is a regular minimal dominating set of an IFHG .
Hence, is regular dominating and independent set of IFHG.
4. Application of the Present Study
Intuitionistic fuzzy graphs have applicability in several fields. In particular, the domination and double domination in IFHG could be applied in decision-making processes. Xing et al. [30] applied A Choquet integral-based interval Type-2 trapezoidal fuzzy multiple attribute group decision-making for sustainable supplier selection. In this work, we used the double domination of IFHG in decision-making for selecting containment zones. For the past two years, the spread of COVID-19 has been an important issue worldwide. Several countries put complete lockdown to control the spread of the epidemic. However, there are several difficulties with the complete lockdown. The lockdown is now imposed in particular areas (Containment zones) with more active COVID-19 positive cases.
The double domination number of IFHG can be applied to determine the containment zones for the epidemic situation like COVID-19. For example, let a city has recorded five active COVID-19 positive cases and if there are eight zones in the city. Now, draw an intuitionistic fuzzy hypergraph by considering the zones as vertices and the travel history of each COVID-19 positive person as hyperedges. The membership value of vertices could be obtained from the combination of data that escalates the rate of spread of the disease, such as population density and unawareness of people about COVID-19 in respective blocks. Further, nonmembership values may be obtained from a combination of data that decrease the spread of COVID-19, such as the administration’s precautionary measures and the awareness of the people about COVID-19.
The membership value of edges could be obtained from the combination of data that escalate the rate of spread of the disease by the respective person. Similarly, nonmembership values may be obtained from a combination of data about the person that decreases the spread of COVID-19. It is easy to find the containment zones by finding that minimal double dominating set of the intuitionistic fuzzy hypergraph. Each vertex in the minimal double dominating set has to be set as a containment zone to control the further spread of COVID-19.
From Figure 12 and Table 12, we notice that all the hyperedges of the IFHG are strong edges, and is the double dominating set of the IFHG. Hence, we can make the respective blocks of the city as the containment zones for controlling the spread of the epidemic.
5. Conclusion
The present analysis discussed domination, double domination, and regular domination in IFHG, which has enormous application in computer science, networking, and chemical and biological engineering. In particular, the domination and double domination in IFHG could be applied in decision-making processes. The double domination number of IFHG can be applied to determine the containment zones for the epidemic situation like COVID-19. Some important definitions are given with examples. Many real field problems can be solved using this technique of double domination, such as transportation problems, social networking problems, and sports modeling. We illustrated an application to determine the containment zones for epidemic situations like COVID-19 with a minimal double dominating set of IFHG. However, a few limitations are there. Calculating actual value, falsity, and indeterminacy from crisp data is challenging to capture. There are no available methods to find such data. Furthermore, the data relating to the travellers who have been wandering across the blocks has not been fully detailed. More real field problems can be solved in future studies through domination, double domination, and regular domination of IFHG.
The derived results are given below:(i)The double dominating set of an IFHG exists only if every vertex in contains at least two other vertices as strong neighbors.(ii)The double dominating set of an IFHG is the set of all vertices that IFGH if all of its vertices are end vertices.(iii)Let H be an IFHG, if the double dominating set of is an independent set of , then it is not an independent set of .(iv)A regular independent set is a regular maximal independent set of an IFHG if it is a regular independent and dominating set of IFHG.(v)A regular dominating set is a regular minimal dominating set of an IFHG if it is a regular dominating and independent set of IFHG.
Nomenclature
| Notation: | Meaning |
| : | Universal set |
| : | Fuzzy set |
| : | Edge set of the graphs |
| : | Vertices set of the graphs |
| : | The membership degree of element |
| : | The nonmembership degree of element |
| : | The membership degree of a vertex |
| : | The nonmembership degree of a vertex |
| : | The membership degree of an edge |
| : | The nonmembership degree of an edge |
| IFHG: | Intuitionistic Fuzzy Hypergraph |
| : | The degree of vertex |
| : | The minimum degree of the IFHG |
| : | The maximum degree of IFHG |
| : | The dominating set of IFHG |
| : | The double dominating set of IFHG |
| : | The regular dominating set of IFHG |
Data Availability
No data were used for this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.