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VG can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed in bringing about satisfactory results. The previous definitions’ restrictions in FGs have made us present new definitions in VGs. A wide range of applications have been attributed to the domination in graph theory for several fields such as facility location problem, school bus routing, modeling biological networks, and coding theory. Therefore, in this research, we study several concepts of domination, such as restrained dominating set (RDS), perfect dominating set (PDS), global restrained dominating set (GRDS), total

Graph theory began its adventure from the well-known “Konigsberg bridge problem.” This problem is frequently believed to have been the beginning of graph theory. In 1739, Euler finally elucidated this problem using graphs. Even though graph theory is an extraordinarily old concept, its growing utilization in operations research, chemistry, genetics, electrical engineering, geography, sociology, and so forth has reserved it fresh. In recent times, graph principle has been utilized in communication system (mobile, internet, etc.), computer layout, and so forth. In graph theory, it is far considered that the nodes, edges, weights, and so on are definite. To be exact, there may be no question concerning the existence of these objects. However, the real world sits on a plethora of uncertainties, indicating that, in some conditions, it is believed that the nodes, edges, and weights may additionally be or may not be certain. For instance, the vehicle travel time or vehicle capacity on a road network may not be identified or known exactly. To embody such graphs, Rosenfeld [

The notion of vague set theory, generalization of Zadeh’s fuzzy set theory, was introduced by Gau and Buehrer [

Domination in VGs theory is one of the most widely used topics in other sciences, including psychology, computer science, nervous systems, artificial intelligence, decision-making theory, and combinations. Although the dominance of FGs has been stated by some researchers, due to the fact that VGs are wider and are more widely used than FGs, it is observed today that they are used in many branches of engineering and medical sciences. Likewise, they have been used in many applications for the formulation and solution of many problems in various areas of science and technology exemplified by computer networks, combinatorial analyses, physics, and so forth. In 1962, Ore [

In this section, some basic concepts of VGs are reviewed to facilitate the next sections.

A graph denotes a pair

An FG has the form of

(see [

(see [

(see [

(see [

(see [

An edge

(see [

(see [

(see [

(see [

(see [

(see [

Notations are shown in Table

Some basic notations.

Notation | Meaning |
---|---|

FG | Fuzzy graph |

VS | Vague set |

Vague graph | |

DS | Dominating set |

RDS | Restrained dominating set |

IDS | Independent dominating set |

IDN | Independent dominating number |

IR-DN | Irredundance dominating number |

K-DS | K-dominating set |

PDS | Perfect dominating set |

PDN | Perfect dominating number |

K-DN | K-dominating number |

T-KDN | Total K-dominating number |

MI-PDS | Minimal perfect dominating set |

T-KDS | Total K-dominating set |

RDN | Restrained dominating number |

GRDN | Global restrained dominating number |

CVG | Complete vague graph |

EDS | Equitable dominating set |

GRDS | Global restrained dominating set |

Let

Let

Consider an example of a T-2DS

It is clear from Figure

Total 2-dominating set of

Let

Consider a VG

RDS of

Let

Consider

Global RD set of

Suppose that

Assume that

Let

We say that a PDS

Consider a VG

VG

Every DS in CVG

Let

The strong product of two VGs

Let

Assume that

The strong product

If a node

Suppose that

The direct product of two VGs

If a node

Direct product of

Consider a VG as in Figure

Let

Consider a VG

It is obvious that the MI-EDS of a VG

Equitable dominating set of

Let

Assume that

Let

To prove that

If

If

If

Let

To prove that each two distinct nodes

If

If

Hence,

One year has passed since the beginning of the coronary heart disease pandemic in the world. During this year, many people have died in all countries and the lives of all people have been affected. During this period, no definitive cure for this disease has been found and many countries, in attempts to develop a corona vaccine to prevent the disease, are highly contagious. China, Russia, India, and the United States are among these countries, and of course Iran has made efforts in this regard. Most vaccines are in the final stages of production and are about to be sacrificed, and many countries have prepurchased several million doses of these vaccines at this stage. Some vaccines are artificially made from antibodies created following disease; and some other viruses have been killed or weakened. The effectiveness of the study population and less side effects are the most important issues in choosing a vaccine. Relations between countries and political issues between them are also factors affecting the type and amount of vaccines purchased. Although it has been said that the whole world should be safe and these vaccines should be given to all countries, the issues mentioned are definitely on the time required to establish comprehensive security in each country will be effective. Therefore, in this paper, we try to discuss the application and importance of domination in the field of medical sciences and discuss the topic in today’s world, namely, the corona vaccine. For this purpose, we consider five countries: Iran, China, USA, India, and Russia. In fact, we want to buy the most effective vaccine for Iran, given the effectiveness of the vaccine and the political relations that exist between this country and other countries. In this vague graph, the nodes representing the countries and edges indicate the extent of political relations and friendship between the two countries.

The vertex of China

VG

After calculating the cardinality of

It is clear that

Domination in FGs theory is one of the most widely discussed topics in other sciences including psychology, computer science, nervous systems, artificial intelligence, and combinations. They have also been utilized in summarizing document and in designing secure systems for electrical grids. Hence, in this paper, we introduced several concepts of domination, such as RDS, PDS, GRDS, EDS, and total K-dominating set in VGs and also investigated their properties by some examples. Finally, we described an application of domination in the field of medical sciences and discussed a topic in today’s world, namely, the coronavirus. In our future work, we will introduce vague incidence graphs and study the concepts of connected perfect dominating set, regular perfect dominating set, inverse perfect dominating set, and independent perfect dominating set on vague incidence graph.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This work was supported by the National Key R&D Program of China (no. 2018YFB1005100).