# Double Domination and Regular Domination in Intuitionistic Fuzzy Hypergraph

Authors: P. Aruna Sri; N. Thamaraikannan; K. Loganathan; Dinesh Kumar Chaudhary
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 setU. The fuzzy set A in U is represented by A=x,µAx:x>0,x∈E,where the functionµA:U⟶0,1 is the membership degree of element x∈U in the fuzzy set A.Definition 2.
For a fixed setU be, an Intuitionistic Fuzzy set (IFS) A in U is of the form A=x,µAx,γAx:x>0,x∈U, where the function µA:U⟶0,1 and γA:U⟶0,1 are determined as the “membership, and non-membership degree of element x∈U respectively” and for every x∈U, (1)0≤µAx+γAx≤1.Definition 3.
An Intuitionistic Fuzzy Graph (IFG) is of the formG=V,E, where (i)
V=v1,v2,…vm such that µ1:V⟶0,1 and γ1:V⟶0,1 denote the degree of membership and nonmembership of the element vi∈V, respectively, and 0 ≤µ1vi+γ1vi≤1, for every vi∈Vi=1,2,…m(ii)
E∈V×V, where such that µ1:V⟶0,1 and γ1:V⟶0,1 are such that(2)µ2vi,vj≤minµ1vi,µ1vj,γ2vi,vj≤maxγ1vi,γ1vj,and 0 ≤µ2x+γ2x≤1 for every vi,vj∈E, where i=1,2…m,j=1,2.…n.
Here, the degree of membership and degree of nonmembership of the vertexvi is denoted by the triple (vi, µ1i,γ1i). The degree of membership and degree of nonmembership of the edge relation eij=vi,vj on V is denoted by the triple eij,µ2ij,γ2ij.Definition 4.
The crisp subset ofX in which all its elements have nonzero membership degree is defined as the support of the fuzzy set A:(3)supppA=x:µAx>0.Definition 5.
An intuitionistic fuzzy hypergraph (IFHG)H is an ordered pair H=V,E, where (i)
V=v1,v2,…vm a finite set of vertices(iii)
E=E1,E2,…Em a finite intuitionistic fuzzy subset of V(iii)
Ej=vi,µjvi,γjvi:µjvi,γjvi≥0 and 0≤µjvi+γjvi≤1j=1,2,…m(iv)
Ej≠Φ, j=1,2,…m(v)
∪jsuppEj=V,i=1,2….m,j=1,2,…n
Here the edgesEj are intuitionistic fuzzy sets. µjviandγjvi denote the degree of membership and degree of nonmembership of the vertex vi to the edge Ej. Thus, the elements of the incidence matrix of intuitionistic fuzzy hypergraph are of the form (vij, µjvi,γjvi), the sets VandE are crisp sets.Definition 6.
An intuitionistic fuzzy hypergraphH′=V′,E′ is said to be an intuitionistic fuzzy subhypergraph (IFHSG) of =V,EifV′⊆VandE′⊆E, that is, µ1i′≤µ1i;γ1i′≥γ1i and µ2ij′≤µ2ij;γ2ij′≥γ2ij for every i,j=1,2,…m.Definition 7.
The orderOH of an intuitionistic fuzzy hypergraph H is the number of vertices of the number of hyperedges is called as the size of the intuitionistic fuzzy hypergraph and is denoted by SH.Definition 8.
The degree of a vertexV in an Intuitionistic fuzzy Hyper graph H=V,E is defined as the sum of the weights of the strong edges incident at V. It is denoted by dhv. The minimum degree of H is δH=mindhv:v∈V. The maximum degree of H is ΔH=maxdhv:v∈V.”Definition 9.
An edgevi,vj of an IFHG is semi–µ-strong IFHG if µ2ij=µ1iµ1j, where, i,j=1,2,…m.Example 1.
From Figure1and Table 1, semi–µ-strong edges are E1E2,E4,E5, E6.Figure 1
Semi–µ-strong edges of IFHG H.Table 1
Degree of membership and nonmembership values for vertices and edges of Figure1.
Verticesviv1v2v3v4v5v6μ1i,γ1i(0.2,0.3)(0.5,0.1)(0.4,0.3)(0.2,0.6)(0.1,0.1)(0.4,0.1)EdgesEJE1E2E3E4E5E6μ2i,γ2i(0.0016,0.0054)(0.008,0.016)(0.8,0.09)(0.08,0.18)(0.1,0.03)(0.2,0.03)Definition 10.
An edgevi,vj of an IFHG is semi–γ-strong IFHG if γ2ij=γ1iγ1j.Example 2.
From Figure2 and Table 2, semi–γ strong edges of H are. E2,E3,Figure 2
Semi–γ-strong edge of IFHG H.Table 2
Degree of membership and nonmembership values for vertices and edges of Figure2.
Verticesviv1v2v3v4v5μ1i,γ1i(0.6,0.1)(04,0.2)(0.1,0.2)(0.3,0.5)0.3,0.1)EdgesEJE1E2E3----μ2i,γ2i(0.03,0.7)(0.036,0.01)(0.0054,0.001)----Definition 11.
An edgevi,vj of an IFHG is strong if µ2ij=µ1iµ1jandγ2ij=γ1iγ1j, where i=1,2,…m, j = 1,2, ..., n.Example 3.
From Figure3 and Table 3, the semi–µ-strong edges of H are E1, E2,E3 and semi-γ-strong edges are E1, E2,E3,E4.
Therefore, the strong edges ofH are E1, E2, and E3.Figure 3
Strong edge of IFHGH.Table 3
Degree of membership and nonmembership values for vertices and edges of Figure3.
Verticesviv1v2v3v4v5μ1i,γ1i(0.03,0.5)(0.1,0.1)(0.5,0.4)(0.7,0.2)(0.1,0.3)EdgesEJE1E2E3E4--μ2ij,γ2ij(0.005,0.012)(0.0007,0.0012)(0.35,0.08)(0.2,0.15)--Definition 12.
Any two subsets of verticesvi,vj of an IFHG are semi–µ-dominating each other if there exist a semi–µ-strong edge vi,vj.Definition 13.
Any two subsets of verticesvi,vj of an IFHG are semi–γ-dominating each other if there exist a semi γ-strong edge vi,vj.Definition 14.
Any two subsets of verticesvi,vj of an IFHG are dominating each other if there exist a strong edge vi,vj.Definition 15.
LetH be an IFHG, a subset Dh of V is called a dominating set of H, if for every v belongs to V-Dh, there is u belongs to Dh, such that u and v are strong neighbors.Example 4.
From Figure4 and Table 4, the strong edges of the IFHG H are E1, E2,E5. The dominating sets of IFHG H are v1,v3,v5,v2,v3,v6.Figure 4
Dominating set of IFHGH.Table 4
Degree of membership and nonmembership values for vertices and edges of Figure4.
Verticesviv1v2v3v4v5v6μ1i,γ1i(0.1,0.3)(0.7,0.1)(0.5,0.4)(0.2,0.3)(0.6,0.3)(0.1,0.1)EdgesEJE1E2E3E4E5--μ2ij,γ2ij(0.006,0.0108)(0.084,0.009)(0.007,0.4)(0.2,0.5)(0.1,0.12)--Definition 16.
An IFHG,H=V,E is complete µ-strong IFHG, if µ2ij=minµ1iµ1j and γ2ij<maxγ1iγ1j, where vi,vj∈V.Definition 17.
An IFHG,H=V,E is complete γ-strong IFHG, if µ2ij<minµ1iµ1j and γ2ij=maxγ1i,γ1j, where vi,vj∈V.Definition 18.
An IFHG,H=V,E is complete IFHG, if µ2ij=minµ1iµ1j) and γ2ij=maxγ1iγ1j, where vi,vj∈V.Definition 19.
LetH be an IFHG, a subset S of V is called independent set of H if µ2ij≠µ1iµ1j and γ2ij≠γ1iγ1j, where i=1,2…m,j=1,2….n.Example 5.
From Figure5 and Table 5, the strong edges of the IFHG H are E1, E3,E4.
The independent sets of the IFHGH are {v1,v6,v7,v2,v7,v6....}Figure 5
Independent setS of IFHG H.Table 5
Degree of membership and nonmembership values for vertices and edges of Figure5.
Verticesviv1v2v3v4v5v6μ1i,γ1i(0.2,0.7)(0.6,0.2)(0.4,0.3)(0.3,0.5)(0.3,0.1)(0.1,0.1)EdgesEJE1E2E3E4----μ2ij,γ2ij(0.0052,0.0105)(0.1,0.5)(0.054,0.01)(0.03,0.05)Definition 20.
An independent setS of an IFHG H is said to be a maximal independent set, if for every vertex v∈V−S, the set S∪v is not an independent set.Definition 21.
Two vertices in an IFHG,H=V,E are said to be independent set if there is no strong edge between them.Definition 22.
A vertexu∈V of an IFHG, H=V,E are isolated vertices if µ2u,v=0 and γ2u,v=0 for all u∈V.(i.e.) Nu=∅.
Thus, the other vertices inH should not be dominated by an isolated vertex.Figure 6
IFHGH.Figure 7
The complementH¯ of IFHG H.Definition 23.
The complement of an IFHG,H=V,E is an IFHG H¯=V¯,E¯, where (i)
V¯=V(ii)
µ¯1i=µ1i and γ¯1i=γ1i for all i=1,2,…m(iii)
µ¯2ij=minµ1i,µ1j−µ2ij and γ¯2ij=minγ1i,γ1j−γ2ijγ1j for all i=1,2,…m, j=1,2...n.Example 6.
From Tables6 and 7, it is found that IFHG in Figure 7 is the complement IFHG of Figure 6.Table 6
Degree of membership and nonmembership values for vertices and edges of Figure6.
Verticesviv1v2v3v4v5μ1i,γ1i(0.3,0.6)(0.3,0.4)(0.5,0.2)(0.7,0.1)(0.1,0.1)EdgesEJE1E2E3E4--μ2ij,γ2ij(0.25,0.002)(0.00315,0.00048)(0.000315,0.000048)(0.009,0.024)Table 7
Degree of membership and nonmembership values for vertices and edges of Figure7.
VerticesV¯iv¯1v¯2v¯3v¯4v¯5μ¯1i,γ¯1i(0.3,0.6)(0.3,0.4)(0.5,0.2)(0.7,0.1)(0.1,0.1)EdgesE¯jE¯1E¯2E¯3E¯4--μ¯2ij,γ¯2ij(0.5,0.58)(0.09,0.36)(0.0085,0.003)(0.792,0.7)Definition 24.
LetH=V,E be an IFHG. A subset D2h of V is a double dominating set of H, if for each vertex in V-D2h is dominated at least by two vertices in D2h is the minimum fuzzy cardinality of all the double dominating set of H is defined as the double domination number of H and denoted by γddhH.Figure 8
Double dominating setD2h of an IFHG.Example 7.
From Figure8 and Table 8, the strong arcs are E1E2, E3,E4, so the double dominating set of the IFHG is D2h =  v2,v6,v4,,v5, where V-D2h =  v1,v6Table 8
Degree of membership and nonmembership values for vertices and edges of Figure8.
Verticesviv1v2v3v4v5v6μ1i,γ1i(0.5,0.4)(0.6,0.4)(0.11,0.5)(0.1,0.2)(0.4,0.4)(0.3,0.5)EdgesEJE1E2E3E4E5--μ2ij,γ2ij(0.006,0.016)(0.004,0.05)(0.03,0.032)(0.04,0.2)(0.026,0.032)--Definition 25.
A vertexV∈H is an end vertex of IFHG, and it has at most one strong neighbor in H.Definition 26.
A vertexVj, j=1,2...n is said to be a cut vertex in IFHG if deleting a vertex Viwherei=1,2...m reduces the strength of the connectedness between some pair of vertices.Definition 27.
Letu be a vertex in an IFHG, H=V,E the set Nu=v∈V/u,vis  a strong  arc is called neighborhood of u.Figure 9
Regular double dominating setDr2h of IFHG H.Definition 28.
LetH=V.E be an IFHG. A set Drh subset of V is called regular intuitionistic fuzzy hyper dominating set if (i)
Every vertex inV−Drh has strong edge to at least one vertex in Drh.(ii)
The hyperdegree of all the vertices inDrh should be same.Definition 29.
LetH=V.E be an IFHG. A set Dr2h subset of V is called regular double dominating set of intuitionistic fuzzy Hyper graph if (i)
Every vertex inV−Dr2h adjacent to at least two vertices in D2h.(ii)
All the vertices inD2h has the same hyperdegree.Example 8.
From Figure9 and Table 9, the strong arcs of the IFHG H are E2E3,E4; hence, the regular double dominating set of H is Dr2h=v1,v2, here Dr2h=v1,v2.Table 9
Degree of membership and nonmembership values for vertices and edges of Figure9.
Verticesviv1v2v3v4μ1i,γ1i(0.5,0.5)(0.7,0.2)(0.1,0.1)(0.2,0.3)EdgesEJE1E2E3E4μ2ij,γ2ij(0.3,0.15)(0.001,0.015)(0.014,0.006)(0.35,0.1)Definition 30.
The minimum cardinality of regular intuitionistic fuzzy hyper dominating set is regular intuitionistic fuzzy hyper domination number, and it is denoted byγrifhH.Definition 31.
LetH be an IFHG. A set Dh subset of V is called minimal regular intuitionistic fuzzy hyper dominating set if (i)
Any subset ofDh is not a regular intuitionistic fuzzy hyper dominating set.(ii)
All the vertices inDh has the same hyperdegree.Definition 32.
An independent set of an IFHG,H=V,E is said to be regular independent intuitionistic fuzzy hyper set if a set Dh is subset of V, (i)
µ2ij≠µ1iµ1j and γ2ij≠γ1iγ1j(ii)
All the vertices inDh has d hyperdegree and d hyperedges.Definition 33.
LetH be an IFHG. A set Dh subset of V is called maximal regular independent intuitionistic fuzzy hyperset if for every vertex v∈V−Dh, the set Dh∪​v 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 inV−D2h contains at least two other vertices as strong neighbors.Proof.
LetD2h is a double dominating set of an IFHG. If there exists a vertex in V−D2h with a single strong neighbor, let it be u and its strong neighbor is v.Case 1.
Ifv∈V−D2h, then u∈V−D2h has no strong neighbor in D2h, this implies that D2h cannot be a double dominating set.Case 2.
IfV∈D2h such that u∈V−D2h has exactly one strong neighbor, again, this implies that D2h cannot be a double dominating set.
We obtain contradiction in both the cases. Hence, there exist at least two strong neighbors for every vertex inV−D2h.Figure 10
IFHGH.Example 9.
From Figure10 and Table 10, the strong arcs of IFHG H are E1E3,E4; hence, D2h=v2,v3,v4,,v5, V−D2h=v1.
v2andv3 has at least two neighbors.Table 10
The values for degree of membership and nonmembership of vertices and edges of Figure10.
Verticesviv1v2v3v4v5μ1i,γ1i((0.1,0.1)(0.3,0.5)(0.1,0.3)(0.5,0.5)(0.7,0.1)EdgesEJE1E2E3E4--μ2ij,γ2ij(0.05,0.15)(0.027,0.015)(0.007,0.003)(0.021,0.005)--Theorem 2.
The double dominating setD of an IFHG H is the set of all vertices V of that IFGH if and only if all of its vertices are end vertices.Proof.
Let all the vertices of IFHGH are end vertices. Since H has only end vertices, all of its vertices have exactly one strong neighbor.
Also from Theorem1, all the vertices in V−D2h should have at least two strong neighbors, which imply that there are no vertices in V−D2h double dominating set.
That is,V−D2h=Φ, Hence, we have V−D2h.Figure 11
IFGHH with only end nodes.Example 10.
From Figure11 and Table 11, the strong arcs are E1E3., and all the vertices of H are end vertices. Hence, the double dominating set of H is V itself.Table 11
Degree of membership and nonmembership values for vertices and edges of Figure11.
Verticesviv1v2v3v4v5μ1i,γ1i(0.3,0.6)(0.2,0.4)(0.7,0.1)(0.1,0.8)(0.1,0.3)EdgesEJE1E2E3E4--μ2ij,γ2ij(0.03,0.48)(0.942,0.624)(0.02,0.12)(0.0014,0.8096)--Theorem 3.
LetH be an IFHG, and if the double dominating set D2h of H is an independent set of H, then it was not an independent set of H¯.Proof.
LetD2h be a double dominating set of H. If D2h is an independent set of H. Let H¯ be a complement of intuitionistic fuzzy hypergraph H. In H¯=V¯,E¯, where V¯=V, µ¯1i=µ1i and γ¯1i=γ1i for all i=1,2,…m, µ¯2ij=minµ1i,µ1j−µ2ij and γ¯2ij=minγ1i,γ1j−γ2ij, for all i,=1,2,…m, j=1,2...n. Here only the values of hyperedges are changed in H¯, which implies most of the adjacent vertices in H¯ have strong neighbors and also different double dominating set exists in H¯.
Therefore, the same double dominating setD2h in H cannot be an independent set of H¯.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.
LetD2h be a regular maximal independent set in an IFHG, then for every u∈V−Drh the set Drh∪u is not an independent set. It is trivial that D2h is regular independent of H. It is enough to prove that D2h Regular dominating set, suppose that D2h is not a regular dominating set, and then there exists at least one vertex v∈V−Drh, such that there is no strong neighbor for v in Drh, which implies that Drh∪v is an independent set of H. This contradicts the fact that Drh is a regular maximal independent set of an IFHG H. Hence, Dh 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.
LetDrh is regular minimal dominating set of H. It is trivial that Drh is regular dominating of H. It is enough to prove that Drh Regular independent set, suppose that Drh is not a regular independent set, then there exists at least one vertex v∈Drh such that at least one vertex in Drh−v has a strong hyper-edge to V, which implies Drh−v is regular dominating set. This contradicts the fact that Drh is a regular minimal dominating set of an IFHG H.
Hence,Drh 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 Figure12 and Table 12, we notice that all the hyperedges of the IFHG are strong edges, and D2h=v2,v5 is the double dominating set of the IFHG. Hence, we can make the respective blocks v2,v5 of the city as the containment zones for controlling the spread of the epidemic.Figure 12
Application of double dominating in IFGH.Table 12
Degree of membership and nonmembership values for vertices and edges of Figure11.
Verticesviv1v2v3v4v5v6v7v8μ1i,γ1i(0.1,0.6)(0.2,0.5)(0.7,0.1)(0.1,0.3)(0.7,0.3)(0.6,0.1)(0.2,0.8)(0.1,0.1)EdgesEJE1E2E3E4E5------μ2ij,γ2ij(0.000588,0.00009)(0.014,0.015)(0.014,0.072)(0.028,0.12)(0.02,0.3)------


## 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 inV−D2h contains at least two other vertices as strong neighbors.(ii)
The double dominating setD2h of an IFHG H is the set of all vertices that IFGH V if all of its vertices are end vertices.(iii)
Let H be an IFHG, if the double dominating setD2h of H is an independent set of H, then it is not an independent set of H¯.(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.



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