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Social networks are represented using graph theory. In this case, individuals in a social network are assumed as nodes. Sometimes institutions or groups are also assumed as nodes. Institutions and such groups are assumed as cluster nodes that contain individuals or simple nodes. Hypergraphs have hyperedges that include more than one node. In this study, cluster hypergraphs are introduced to generalize the concept of hypergraphs, where cluster nodes are allowed. Sometimes competitions in the real world are done as groups. Cluster hypergraphs are used to represent such kinds of competitions. Competition cluster hypergraphs of semidirected graphs (a special type of mixed graphs called semidirected graphs, where the directed and undirected edges both are allowed) are introduced, and related properties are discussed. To define competition cluster hypergraphs, a few properties of semidirected graphs are established. Some associated terms on semidirected graphs are studied. At last, a numerical application is illustrated.

Cohen introduced the concept of a competition graph [

There are various research works on competition graphs. Roberts et al. studied that any graph with isolated vertices is the competition graph [

A hypergraph is a generalization of a graph in which any subset of a vertex set is an edge rather than two vertex sets. Especially, Berge [

Let

The pair

Burosch and Ceccherini [

Mixed graphs represent social networks accurately as there may be directed edges and undirected edges. Directed edges are the indication of influences, dominations, or followers while undirected edges are the indication of the opposite cases where individuals are connected but without following each other. The study of mixed graphs [

The definition of a semidirected graph is given as follows.

(see [

The preliminary terms are not defined in the literature. These terms are defined as follows. Incidence number which is related to the degree of a node is defined in the following.

The degree of a vertex

The degree of a vertex

A related property has been developed as follows.

A semidirected graph.

The sum of the incidence numbers in a semidirected graph is always even.

Let

Since one undirected edge contributes two degrees, then

Hence,

The definitions of complete semidirected graphs are introduced in the following.

If there exist all three types of connections, i.e., out-directed edges, in-directed edges, and undirected edges, between every pair of vertices, then the graph is called a complete semidirected graph.

Without having all three types of edges, sometimes graphs may be completely depending on the incidence number and connections between every pair of vertices.

A semidirected graph is said to be a complete-incidence semidirected graph if every pair of vertices is connected by at least one edge (undirected or directed), and the incidence number of all vertices is equal.

Since, as shown in Figure

An example of a complete-incidence semidirected graph.

Neighbourhood, out-neighbourhood, and in-neighbourhood of a vertex

The maximal out-neighbourhood vertex set is an out-neighbourhood of a vertex which is not contained in other out-neighbourhood of any vertices.

The maximal in-neighbourhood vertex set is an in-neighbourhood of a vertex which is not contained in other in-neighbourhood of any vertices.

In Figure

Thus,

A semidirected graph.

The

In Figure

The

(i) Vertex set of

(ii) Edge set of

In Figure

2-step semidirected graph of Figure

Let

For each element

Then,

The

Let

For each element

Then,

In Figure

A cluster hypergraph.

Virtual representation of Figure

The vertex set of a cluster hypergraph may contain a group of people/individuals in a network as a node (cluster node) while all the people in the network are assumed as simple nodes. This concept is helpful to assume any organisation or group as nodes in any network. Also, it is assumed that each node inside a cluster node is automatically connected to the cluster node, but these inside nodes may not be connected to each other.

The definition of a cluster hypergraph includes the concept of the multi-hyperedge set. Thus, the repetition of elements in

In a virtual representation (Figure

Depending on the cluster node sizes and their edges, cluster hypergraphs are classified into different categories. To classify, maximal nodes are to be defined. The maximal nodes are those nodes which are not contained in any other cluster nodes. The elements of

Virtual representations of CCCH (2-cluster hypergraphs have been assumed here).

A cluster hypergraph is said to be a

A (2,3)-uniform cluster hypergraph and its virtual representation.

A cluster hypergraph is said to be cluster connected cluster hypergraphs (CCCH) if there are edges only connecting the maximal cluster nodes. In that case, maximal cluster nodes are automatically connected to its internal nodes. In Figure

Let

A complete CCCH and its virtual representation.

A complete uniform CCCH and its virtual representation.

A complete

Let us consider a complete

Competition cluster hypergraphs of semidirected graphs are defined where adjacent vertices by undirected edges form a cluster. If these adjacent vertices have common out-directed neighbours, then these vertices are also adjacent in competition cluster hypergraphs. The formal definition is given as follows.

Let

Let us consider a semidirected graph, as shown in Figure

A competition cluster hypergraph. (a) A semidirected graph

Let

Let

Let

Let

Let

(

(

Let

(double competition cluster hypergraphs). Let

A semidirected graph is shown in Figure

Double competition cluster hypergraphs. (a) A semidirected graph. (b) Double competition cluster hypergraph.

(

Let us consider a semidirected graph, as shown in Figure

2-step competition cluster hypergraph.

A cluster hypergraph with two isolated nodes.

A cluster hypergraph with one isolated node.

A node in a cluster hypergraph is called an isolated node if there exist no edge to the node from the nodes of the graph. There are two types of isolated nodes.

Consider a cluster hypergraph

Consider a cluster hypergraph (Figure

A cluster hypergraph

Corresponding semidirected graph of

The steps to find the competition number of a cluster hypergraph are given as follows:

Competition number of a

Consider a

Let G be a semidirected graph and G_{m} be the m-step semi directed graph of G, then

Since G is a semi‐directed graph and G_{m} is the m‐step semi‐directed graph of G. Then vertex set of both graphs is the same. Let

In a semidirected graph

Since

This study developed basic terminologies of mixed graphs. Some properties have been discussed. Another notion is that the cluster hypergraphs are also introduced. And finally, competition cluster hypergraphs of such semidirected graphs have been described. These types of competition graphs can be used to represent group competitions. A network is considered for COVID-19-affected areas as follows to show such a competition.

Affected places are assumed as nodes of a semidirected graph along with the source node as “COVID-19.” Another different topic, major carbon emission countries are added in this network along with a source node as “carbon emission.” Undirected edges connect the regions which are affected by COVID-19 through other countries. For Tables

Top six COVID-19-affected countries (collected from Wikipedia dated 15 March 2020).

Country | Total affected | Date of the first case | Affected from |
---|---|---|---|

China | 81,218 | 31 December 2019 | |

Italy | 69,176 | 31 January 2020 | China |

USA | 54,935 | 19 January 2020 | China |

Spain | 42,058 | 31 January 2020 | China |

Germany | 33,593 | 25 February 2020 | Italy |

Iran | 24,811 | 19 February 2020 | China |

Top 20 countries of carbon emissions (collected from Wikipedia dated 15 March 2020).

Country | Total emissions | Per capita emissions |
---|---|---|

China | 9.04 Bn | 6.59 |

United States | 5.00 Bn | 15.53 |

India | 2.07 Bn | 1.58 |

Russia | 1.47 Bn | 10.19 |

Japan | 1.14 Bn | 8.99 |

Germany | 729.77 Mn | 8.93 |

South Korea | 585.99 Mn | 11.58 |

Iran | 552.40 Mn | 6.98 |

Canada | 549.23 Mn | 15.32 |

Saudi Arabia | 531.46 Mn | 16.85 |

Brazil | 450.79 Mn | 2.17 |

Mexico | 442.31 Mn | 3.66 |

Indonesia | 441.91 Mn | 1.72 |

South Africa | 427.57 Mn | 7.77 |

United Kingdom | 389.75 Mn | 5.99 |

Australia | 380.93 Mn | 15.83 |

Italy | 330.75 Mn | 5.45 |

Turkey | 317.22 Mn | 4.1 |

France | 290.49 Mn | 4.37 |

Poland | 282.40 Mn | 7.34 |

A semidirected graph.

Competition cluster hypergraphs of Figure

The step-by-step process to find out the competition cluster hypergraphs is given as follows:

This small illustration can be extended to large networks for the detection of competition among different species, different countries, different cultures, etc., in several networks. Few related terms like

A few properties of semidirected graphs have been established

Cluster hypergraphs are introduced

Competition cluster hypergraphs are defined

Various features of competition cluster hypergraphs have been studied

An application/numerical example of COVID-19 networks by using competition cluster hypergraphs has been given

The data, provided in the article, were collected from Wikipedia and

The authors declare that they have no conflicts of interest.