# A Mathematical Approach on Representation of Competitions: Competition Cluster Hypergraphs

Authors: Sovan Samanta; G. Muhiuddin; Abdulaziz M. Alanazi; Kousik Das
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

Cohen introduced the concept of a competition graph [1] with application in an ecosystem which was related to the competition among species in a food web. If two species have at least one common prey, then there is a competition between them. Let G⟶=V,E⟶ be a digraph, which corresponds to a food web. A vertex x∈V represents a species in the food web, and an arc x,s⟶∈E⟶ means x preys on the species s. The competition graph CG⟶ of a digraph G⟶ is an undirected graph G=V,E which has the same vertex set and has an edge between two distinct vertices x,y∈V if there exists a vertex s∈V and arcs x,s⟶,y,s⟶∈E⟶.There are various research works on competition graphs. Roberts et al. studied that any graph with isolated vertices is the competition graph [2, 3], and the minimum number of such vertices is called the competition number. Opsut discussed the computation of competition numbers [4] of a graph. Kim et al. introduced the p-competition graph [5] and also the p-competition number [6]. Brigham et al. introduced the ∅−tolerance graph as a generalization of p-competition [7]. Cho and Kim studied competition numbers [8] of a graph having one hole. Li and Chang proposed about a competition graph [9] with h holes. Factor and Merz introduced the 1,2-step competition graph [10] of a tournament and extended the 1,2-step competition graph. But group competition along with the individual competition is not considered in these papers. This study developed the representations of an individual and group competition by cluster hypergraphs.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 [11, 12] introduced hypergraphs as a generalization of graph theory as follows.LetX=x1,x2,…,xn be a finite set and let E=e1,e2,…,em be a family of subsets of X such that(1)ei≠ϕ,i=1,2,…,m,∪i=1mei=X.The pairX,E is called a hypergraph with vertex set X and hyperedge set E. The elements x1,x2,…,xn of X are vertices of hypergraph H, and the sets e1,e2,…,em are hyperedges of hypergraph H.Burosch and Ceccherini [13] characterised cube hypergraphs, where each hyperedge contains three vertices. Bergen et al. [14] studied infection in hypergraphs. There are various research works on hypergraphs [15–29]. Sonntag and Teichert [30] defined hypertrees, and they extended the notion to competition hypergraphs [31] in another paper in 2004. But, group competition was ignored in the paper. Cluster hypergraphs have been introduced here to capture the notion of a cluster node.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 [32, 33] was started from 1970 where the edges are directed or undirected as follows: Adiga et al. studied the adjacency matrix [34] of mixed graphs. There are various research works related to matrices [35] and isomorphism [36] on mixed graphs. There are many applications of mixed graphs on social networks. For example, Facebook networks [37] allow mixed direction when if at least one friend follows another along with their friendship, then there are directed edges (for following) along with undirected edges (for friendship). In this sense, considering the mixed graph where the edges both directed and undirected are considered as semidirected graphs, we introduced the representation of competition of semidirected graphs as a special type of hypergraphs, called the competition cluster hypergraph where the nodes may be a cluster or a simple node. This study has the following contributions. In Section 2, semidirected graphs and their properties have been developed. In Section 3, cluster hypergraphs and related notions are introduced. The competition cluster hypergraphs and their classifications are analyzed in Section 4. At last, an application and conclusions are drawn in Section 5. Throughout this paper, mixed graphs and semidirected graphs are used synonymously.


## 2. Semidirected Graph

The definition of a semidirected graph is given as follows.Definition 1.
(see [32]). Let V be a nonempty set of elements, called vertices or nodes. Also, let E=E1∪E2⟶ where E1⊂V×V is a set of unordered pairs of vertices, i.e., E1=u,vu,v∈V, called a set of undirected edges, and E2⟶⊂V×V is a set of ordered pair of vertices, E2⟶=a,b⟶a,b∈V, called a set of directed edges. Here, G=V,E1,E2⟶ is said to be a semidirected graph.
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.Definition 2.
The degree of a vertexu is denoted as a triplet Du=du,d+u,d−u where du is the number of all incident edges of u in E1, d+u is the number of out-directed edges of E2⟶ from the vertex u, and d−u is the number of in-directed edges of E2⟶ towards the vertex u. Now, the incidence number of a vertex u is denoted as inu and defined inu=du+d+u−d−u.Example 1.
The degree of a vertexa of the graph, as shown in Figure 1, is given as da=1,1,2. Now, the incidence number is ina=1+1−2=0.
A related property has been developed as follows.Figure 1
A semidirected graph.Theorem 1.
The sum of the incidence numbers in a semidirected graph is always even.Proof.
LetG=V,E1,E2⟶ be a semidirected graph with n vertices. If inu, du, d+u, and d−u denote the incidence number, degree, in-degree, and out-degree of a vertex u, respectively, then invi=dvi+d+vi−d−vi and hence(2)∑i=1ninvi=∑i=1ndvi+∑i=1nd+vi−∑i=1nd−vi.
Since one undirected edge contributes two degrees, then∑i=1ndvi=even number. Also, if there is a directed edge in the graph, then it is a source of one vertex and also a sink of another vertex, thus ∑i=1nd+vi=∑i=1nd−vi.
Hence,∑i=1ninvi = even number is true.
The definitions of complete semidirected graphs are introduced in the following.Definition 3.
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.Definition 4.
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.Example 2.
Since, as shown in Figure2, there are connections (undirected or directed) between every pair of vertices and the incidence number of each of the vertices is 3, it is a complete-incidence semidirected graph.Figure 2
An example of a complete-incidence semidirected graph.Definition 5.
Neighbourhood, out-neighbourhood, and in-neighbourhood of a vertexu in a semidirected graph G=V,E1,E2⟶ are denoted as Nu,N+u,andN−u and defined as follows:(3)Nu=v∈Vu,v∈E1,N+u=v∈Vu,v⟶∈E2⟶,N−u=v∈Vv,u⟶∈E2⟶.
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.Example 3.
In Figure3,(4)Nd=e,N+d=a,b,c,N+e=b,c,N−d=a,b,N−e=a,b,c.
Thus,N+d is a maximal out-neighbourhood set and N−e is a maximal in-neighbourhood set of the semidirected graph assumed in Figure 3.Figure 3
A semidirected graph.Definition 6.
Them−stepneighbourhood, m−step out-neighbourhood, and m−step in-neighbourhood of a vertex u in a semidirected graph G=V,E1,E2⟶ are denoted as Nmu,Nm+u,andNm−u and defined as follows:(5)Nmu=vm∈V:for all paths such thatu−v1−v2−⋯−vm,Nm+u=vm∈V:for all paths such thatu⟶v1⟶v2⟶⋯⟶vm,Nm−u=vm∈V:for all paths such thatu⟵v1⟵v2⟵⋯⟵vm.Example 4.
In Figure3, N2d=∅, N2+d=e, and N2−d=e.Definition 7.
Them-step semidirected graph Gm=Vm,E1m,E2m⟶ of a semidirected graph G=V,E1,E2⟶ is defined as follows:
(i) Vertex set ofGm is Vm=V
(ii) Edge set ofGm is E1m=u,vm:vm∈Nmu, E⟶2m=u,vm⟶:vm∈Nm+uorNm−uExample 5.
In Figure4, the corresponding 2-step semidirected graph of Figure 3 is shown.Figure 4
2-step semidirected graph of Figure3.


## 3. Cluster Hypergraphs

Definition 8.
LetX be a nonempty set and VX be a subset of PX such that ϕ∉VX and X⊂VX. Now, E be a multiset whose elements belong to PPX such that(i)
ϕ∉E(ii)
For each elemente∈E, there exists at least one element v∈VX such that v∈e
Then,G=VX,E is said to be cluster hypergraph where VX is said to be the vertex set and E is said to be the multi-hyperedge set.
Thek-cluster hypergraph is defined as follows.Definition 9.
LetX be a nonempty set and VX be a subset of PkX, k=1,2,3,… such that ϕ∉VX and X⊂VX. Now, E be a multiset whose elements belong to PVX such that(i)
ϕ∉E(ii)
For each elemente∈E, there exists at least one element v∈VX such that v∈e
Then,G=VX,E is said to be k-cluster hypergraph where VX is said to be the vertex set and E is said to be the multi-hyperedge set. Generally, for k=1, 1-cluster hypergraphs are assumed as cluster hypergraphs.Example 6.
In Figure5, X=A,B,C,E,F,G, VX=A,B,C,D=B,C,E,F,G,H=F,G, and E=A,B,A,E,B,C,B,B,C,C,F,G,F,G,F,F,G,G,F,G,B,C. It can be easily verified that for each element e in E, there exists an element v∈VX such that v∈e. For example, for F,G,FinE, there exists an element(6)F,G∈VX,such thatF,G∈F,G,F.Figure 5
A cluster hypergraph.Figure 6
Virtual representation of Figure5.Remark 1.
(i)
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.(ii)
The definition of a cluster hypergraph includes the concept of the multi-hyperedge set. Thus, the repetition of elements inE is allowed.(iii)
In a virtual representation (Figure6) of any cluster hypergraph, the cluster nodes are assumed as separate nodes, and the connections to the inside nodes are shown in the representation.


### 3.1. Types of Cluster Hypergraphs

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 ofX are termed as simple nodes. A simple node may be termed as a maximal node if it does not belong to any other nodes. For example, in Figure 7, the node f is a simple as well as a maximal node, but the node a,b,c,d is a maximal node.Figure 7
Virtual representations of CCCH (2-cluster hypergraphs have been assumed here).


#### 3.1.1. Uniform Cluster Hypergraphs

A cluster hypergraph is said to be am,n-uniform cluster hypergraph if every edge of the hypergraph contains exactly m nodes and each maximal node contains n simple nodes. In Figure 8, a 2,3-uniform cluster hypergraph is shown.Figure 8
A (2,3)-uniform cluster hypergraph and its virtual representation.


#### 3.1.2. Cluster Connected Cluster Hypergraphs (CCCH)

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 Figure7, a CCCH is shown.


#### 3.1.3. Completeness Property of Cluster Hypergraphs

LetX be a nonempty set containing n elements. A cluster hypergraph G=V,E on X contains maximum PX−ϕ number of vertices, i.e., V=2n−1, and for the complete cluster hypergraph, the number of edges is PPX−ϕ−ϕ=22n−1−1. The completeness properties of different types of cluster graphs are discussed as follows.(1) Complete CCCH. A complete CCCH is a cluster hypergraph where any two maximal nodes are connected by an edge. An example of a complete CCCH is shown in Figure 9.Figure 9
A complete CCCH and its virtual representation.(2) Complete Uniform Cluster Hypergraph. A complete uniform cluster hypergraph is a m,n-uniform cluster hypergraph where an edge connects any two maximal nodes and an edge connects any two simple nodes within cluster nodes. An example of a complete (2,3)-uniform cluster hypergraph is shown in Figure 10.Figure 10
A complete uniform CCCH and its virtual representation.Theorem 2.
A completem,n-uniform cluster hypergraph having x cluster nodes contains x×Cmn+Cmx edges.Proof.
Let us consider a completem,n-uniform cluster hypergraph having x cluster nodes. The hypergraph has x cluster nodes containing n nodes per cluster. Thus, the total number of simple nodes is x×n. Also, the graph is complete. Therefore, each edge contains exactly m nodes. Hence, the total number of edges per cluster is Cmn. Also, the total number of edges among x clusters is Cmx. Thus, the number of edges in the complete m,n-uniform cluster hypergraph having x cluster nodes is x×Cmn+Cmx.
Note. A m,n-uniform complete CCCH having x cluster nodes contains Cmx and x×n simple nodes.


## 3.1. Types of Cluster Hypergraphs

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 ofX are termed as simple nodes. A simple node may be termed as a maximal node if it does not belong to any other nodes. For example, in Figure 7, the node f is a simple as well as a maximal node, but the node a,b,c,d is a maximal node.Figure 7
Virtual representations of CCCH (2-cluster hypergraphs have been assumed here).


### 3.1.1. Uniform Cluster Hypergraphs

A cluster hypergraph is said to be am,n-uniform cluster hypergraph if every edge of the hypergraph contains exactly m nodes and each maximal node contains n simple nodes. In Figure 8, a 2,3-uniform cluster hypergraph is shown.Figure 8
A (2,3)-uniform cluster hypergraph and its virtual representation.


### 3.1.2. Cluster Connected Cluster Hypergraphs (CCCH)

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 Figure7, a CCCH is shown.


### 3.1.3. Completeness Property of Cluster Hypergraphs

LetX be a nonempty set containing n elements. A cluster hypergraph G=V,E on X contains maximum PX−ϕ number of vertices, i.e., V=2n−1, and for the complete cluster hypergraph, the number of edges is PPX−ϕ−ϕ=22n−1−1. The completeness properties of different types of cluster graphs are discussed as follows.(1) Complete CCCH. A complete CCCH is a cluster hypergraph where any two maximal nodes are connected by an edge. An example of a complete CCCH is shown in Figure 9.Figure 9
A complete CCCH and its virtual representation.(2) Complete Uniform Cluster Hypergraph. A complete uniform cluster hypergraph is a m,n-uniform cluster hypergraph where an edge connects any two maximal nodes and an edge connects any two simple nodes within cluster nodes. An example of a complete (2,3)-uniform cluster hypergraph is shown in Figure 10.Figure 10
A complete uniform CCCH and its virtual representation.Theorem 2.
A completem,n-uniform cluster hypergraph having x cluster nodes contains x×Cmn+Cmx edges.Proof.
Let us consider a completem,n-uniform cluster hypergraph having x cluster nodes. The hypergraph has x cluster nodes containing n nodes per cluster. Thus, the total number of simple nodes is x×n. Also, the graph is complete. Therefore, each edge contains exactly m nodes. Hence, the total number of edges per cluster is Cmn. Also, the total number of edges among x clusters is Cmx. Thus, the number of edges in the complete m,n-uniform cluster hypergraph having x cluster nodes is x×Cmn+Cmx.
Note. A m,n-uniform complete CCCH having x cluster nodes contains Cmx and x×n simple nodes.


## 3.1.1. Uniform Cluster Hypergraphs

A cluster hypergraph is said to be am,n-uniform cluster hypergraph if every edge of the hypergraph contains exactly m nodes and each maximal node contains n simple nodes. In Figure 8, a 2,3-uniform cluster hypergraph is shown.Figure 8
A (2,3)-uniform cluster hypergraph and its virtual representation.


## 3.1.2. Cluster Connected Cluster Hypergraphs (CCCH)

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 Figure7, a CCCH is shown.


## 3.1.3. Completeness Property of Cluster Hypergraphs

LetX be a nonempty set containing n elements. A cluster hypergraph G=V,E on X contains maximum PX−ϕ number of vertices, i.e., V=2n−1, and for the complete cluster hypergraph, the number of edges is PPX−ϕ−ϕ=22n−1−1. The completeness properties of different types of cluster graphs are discussed as follows.(1) Complete CCCH. A complete CCCH is a cluster hypergraph where any two maximal nodes are connected by an edge. An example of a complete CCCH is shown in Figure 9.Figure 9
A complete CCCH and its virtual representation.(2) Complete Uniform Cluster Hypergraph. A complete uniform cluster hypergraph is a m,n-uniform cluster hypergraph where an edge connects any two maximal nodes and an edge connects any two simple nodes within cluster nodes. An example of a complete (2,3)-uniform cluster hypergraph is shown in Figure 10.Figure 10
A complete uniform CCCH and its virtual representation.Theorem 2.
A completem,n-uniform cluster hypergraph having x cluster nodes contains x×Cmn+Cmx edges.Proof.
Let us consider a completem,n-uniform cluster hypergraph having x cluster nodes. The hypergraph has x cluster nodes containing n nodes per cluster. Thus, the total number of simple nodes is x×n. Also, the graph is complete. Therefore, each edge contains exactly m nodes. Hence, the total number of edges per cluster is Cmn. Also, the total number of edges among x clusters is Cmx. Thus, the number of edges in the complete m,n-uniform cluster hypergraph having x cluster nodes is x×Cmn+Cmx.
Note. A m,n-uniform complete CCCH having x cluster nodes contains Cmx and x×n simple nodes.


## 4. Competition Cluster Hypergraphs

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.Definition 10.
LetG=X,E1,E2⟶ be a semidirected graph where X is a nonempty vertex set, E1 is the set of undirected edges, and E2⟶ is the set of directed edges. Now, the competition cluster hypergraph of G is denoted as CG=VX,E where VX⊂PX is the vertex set of CG such that X⊂VX, xi,xj∈VX if xi,xj∈E1 and x1,x2,…,xm∈VX if x1,x2,…,xm forms a maximal clique in G and E which is the hyperedge set if there exists an edge containing vertices x1,x2,…,xm if N+x1∩N+x2∩⋯∩N+xm≠ϕ and m=2,…,X.Example 7.
Let us consider a semidirected graph, as shown in Figure11(a). The corresponding competition cluster hypergraph is shown in Figure 11(b). In the semidirected graph, X=a,b,c,d,e,f, and also e,f,c forms a clique. So, the vertex set of the corresponding competition hypergraph is VX=X∪e,f,c. Now, N+a∩N+b=a and N+a∩N+b∩N+c=d, Thus, the edge set of CGisE=ab,abc.Figure 11
A competition cluster hypergraph. (a) A semidirected graphG. (b) Competition cluster hypergraphs of G.
(a)(b)Proposition 1.
LetG=X,E1,E⟶2 be a semidirected graph and the corresponding competition cluster hypergraph of G be CG=VX,E. The number of edges in CG is equal to the number of a maximal in-degree set of vertices in G with cardinality greater than one.Proof.
LetG=X,E1,E⟶2 be a semidirected graph where X is a nonempty vertex set and the corresponding competition cluster hypergraph of G be CG=VX,E. In competition cluster hypergraphs, an edge exists between two vertices xandy if they have a common vertex (or vertices). Along with this, if it is found that the third vertex, say z, has the same common vertex, then that edge will contain all the three vertices x,y, and z and so on. Thus, the number of edges in CG is equal to the number of a maximal in-degree set of vertices in G with cardinality greater than one.Remark 2.
LetG=X,E1,E⟶2 be a semidirected graph and the corresponding competition cluster hypergraph of G be CG=VX,E. The number of cluster nodes in CG is equal to the number of cliques by undirected edges in G.Theorem 3.
LetG=X,E1,E⟶2 be a semidirected graph and the corresponding competition cluster hypergraph of G be CG=VX,E. The number of maximal nodes in CG is equal to a number of nodes which are not adjacent to other vertices by undirected edges in G + (the number of undirected edges which are not part of any cliques in G) + (number of maximal cliques in G).Proof.
LetG=X,E1,E⟶2 be a semidirected graph where X is a nonempty vertex set and the corresponding competition cluster hypergraph of G be CG=VX,E.Case 1.
(E1=ϕ): if the graph G has no undirected edges, then every node in CG is a simple node. Hence, the statement is obvious.Case 2.
(E1≠ϕ): in this case, G contains undirected edges. Those edges may construct maximal cliques or simple undirected edges. One maximal clique in G will correspond to one cluster node in CG. The undirected edges which are not in any cluster also correspond to cluster nodes containing two simple nodes. Hence, the number of maximal nodes in CG is equal to the number of nodes which are not adjacent to other vertices by undirected edges in G + (the number of undirected edges which are not part of any cliques in G) + (number of maximal cliques in G).Corollary 1.
LetG=X,E1,E2⟶ be a semidirected graph and the corresponding competition cluster hypergraph of G be CG=VX,E. All cliques in G correspond to maximal nodes if the cliques are disjoint completely, i.e., no vertex and edges are common.Definition 11.
(double competition cluster hypergraphs). LetG=X,E1,E2⟶ be a semidirected graph where X is a nonempty vertex set, E1 is the set of undirected edges, and E2⟶ is the set of directed edges. Now, the competition cluster hypergraph of G is denoted as CG=VX,E where VX⊂PX is the vertex set of CG such that X⊂VX, xi,xj∈VX if xi,xj∈E1 and x1,x2,…,xm∈VX if x1,x2,…,xm forms a maximal clique in G and E which is the edge set if there exists an edge containing vertices x1,x2,…,xm if N+x1∩N+x2∩⋯∩N+xm≠ϕ and m=2,…,X.Example 8.
A semidirected graph is shown in Figure12(a). The corresponding double competition cluster hypergraph is shown in Figure 12(b).Figure 12
Double competition cluster hypergraphs. (a) A semidirected graph. (b) Double competition cluster hypergraph.
(a)(b)Definition 12.
(m−step competition cluster hypergraphs). Let G=X,E1,E2⟶ be a semidirected graph where X is a nonempty vertex set, E1 is the set of undirected edges, and E2⟶ is the set of directed edges. Now, the m-step competition cluster hypergraph of G is denoted as CmG=VX,E where VX⊂PX is the vertex set of CmG such that X⊂VX, xi,xj∈VX if xi,xj∈E1 and x1,x2,…,xn∈VX if x1,x2,…,xn forms a maximal clique in G and E which is the hyperedge set if there exists an edge containing vertices x1,x2,…,xn if Nm+x1∩Nm+x2∩⋯∩Nm+xn≠ϕ and n=2,…,X.Example 9.
Let us consider a semidirected graph, as shown in Figure3. The corresponding 2-step competition cluster hypergraph CmG is shown in Figure 13. In the semidirected graph, X=a,b,c,d,e, and also d,e∈E1. So, the vertex set of the corresponding 2-step competition hypergraph is VX=X∪d,e. Now, N2+a∩N2+c=b. Thus, the edge set of CmGisE=ac.Figure 13
2-step competition cluster hypergraph.Figure 14
A cluster hypergraph with two isolated nodes.Figure 15
A cluster hypergraph with one isolated node.Definition 13.
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.(i)
Isolated Maximal Node. A maximal node (simple or cluster) is called a maximal isolated node if it has no incident edges, i.e., it is isolated from all other maximal nodes of the cluster hypergraph. Consider a cluster hypergraph as shown in Figure 14. We observe that k,D=g,h are isolated maximal nodes.(ii)
Isolated Node in a Cluster. A simple node may be isolated within a cluster node of a cluster hypergraph. Consider a cluster hypergraph as shown in Figure 15. We observe that only node c is an isolated node in the cluster F=c,b of the cluster hypergraph.Definition 14.
Consider a cluster hypergraphG. Then, the competition number k of G is the minimum number of k maximal isolated nodes with G which forms the competition graph of a semidirected graph.Example 10.
Consider a cluster hypergraph (Figure16) G=VX,E where VX=a,b,a,b,c and E=c,b. Then, the corresponding semidirected graph (Figure 17) is drawn. There is no isolated node. Hence, the competition number of G is 0.Figure 16
A cluster hypergraphG.Figure 17
Corresponding semidirected graph ofG.Algorithm 1.
The steps to find the competition number of a cluster hypergraph are given as follows:Step 1. Consider a cluster hypergraph GStep 2. Draw the required directed or undirected edges to fit a corresponding semidirected graph G′ from GStep 3. The extra nodes if needed to get CG′=G may be taken as isolated nodesStep 4. The minimum number of maximal isolated nodes will be the competition number of GProposition 2.
Competition number of a2,2−uniform cluster hypergraph is zero.Proof.
Consider a2,2−uniform cluster hypergraph and draw the virtual presentation of it. Then, draw the bidirected lines between all nodes. Then, we observe that the semidirected graph is obtained whose corresponding competition cluster hypergraph is the 2,2−uniform cluster hypergraph. Hence, there is no isolated node in the graph, and the competition number is zero. This completes the proof.Theorem 4.
Let G be a semidirected graph and Gm be the m-step semi directed graph of G, then CGm=CmG.Proof.
Since G is a semi‐directed graph and Gm is the m‐step semi‐directed graph of G. Then vertex set of both graphs is the same. Let u,v∈CGm. Then, there exist edges u,x1⟶, v,x1⟶; u,x2⟶, v,x2⟶;…, u,xn⟶; v,xn⟶ for some integer n. Now, N+u∩N+v = x1,x2,…,xn. Since an edge u,x1⟶∈Gm implies there exists a path of length m from u to x1 and similar for v,x1⟶∈Gm; hence u,v∈CmG. Similarly, if an edge x,y∈CmG then it implies x,y∈CGm. Therefore, CGm=CmG.Proposition 3.
In a semidirected graphG=V,E1,E2⟶, if m>V, then CmG is a null cluster hypergraph.Proof.
SinceCmG is an m-step competition cluster hypergraph of G and if m>V, then there does not exist a path of length m in G. Hence, Nm+u=Nm+v=∅andNm+u∩Nm+v=∅ for all u,v∈G. Thus, there does not exist any edge in CmG. Therefore, CmG is a null cluster hypergraph.


## 5. Application and Conclusions

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 Tables1 and 2, the corresponding semidirected graph is shown in Figure 12, and its corresponding competition cluster hypergraph is shown in Figure 13.Table 1
Top six COVID-19-affected countries (collected from Wikipedia dated 15 March 2020).
CountryTotal affectedDate of the first caseAffected fromChina81,21831 December 2019Italy69,17631 January 2020ChinaUSA54,93519 January 2020ChinaSpain42,05831 January 2020ChinaGermany33,59325 February 2020ItalyIran24,81119 February 2020ChinaTable 2
Top 20 countries of carbon emissions (collected from Wikipedia dated 15 March 2020).
CountryTotal emissionsPer capita emissionsChina9.04 Bn6.59United States5.00 Bn15.53India2.07 Bn1.58Russia1.47 Bn10.19Japan1.14 Bn8.99Germany729.77 Mn8.93South Korea585.99 Mn11.58Iran552.40 Mn6.98Canada549.23 Mn15.32Saudi Arabia531.46 Mn16.85Brazil450.79 Mn2.17Mexico442.31 Mn3.66Indonesia441.91 Mn1.72South Africa427.57 Mn7.77United Kingdom389.75 Mn5.99Australia380.93 Mn15.83Italy330.75 Mn5.45Turkey317.22 Mn4.1France290.49 Mn4.37Poland282.40 Mn7.34Figure 18
A semidirected graph.Figure 19
Competition cluster hypergraphs of Figure18.The step-by-step process to find out the competition cluster hypergraphs is given as follows:Step 1. Construction of Semidirected Graphs. Six highest COVID-19-affected countries have been assumed as nodes along with two fictitious nodes COVID-19 and CO2 emissions for a small semidirected graph. All the assumed countries are affected by COVID-19 and CO2 emissions. Thus, there will be direct edges from COVID-19 and CO2 emissions to all the nodes. If one country is affected by others, then there will be undirected edges (see Table 1 and Figure 18).Step 2. Resultant Competition Hypergraphs. In the resultant competition hypergraphs, cliques will form cluster nodes. China-Italy, China-Spain, Italy-Germany, China-USA, and China-Iran are cluster nodes for this case (Figure 19). Between two nodes, there will be edges if the nodes have common out-neighbourhoods in semidirected graphs. Hence, the nodes COVID-19 and CO2 emissions will have one edge.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 likem-step competition cluster hypergraphs and competition numbers have been analyzed with proper examples. This study will be a backbone for a new branch of hypergraphs and cluster hypergraphs along with competition cluster hypergraphs of semidirected graphs. Along with the theoretical developments, these theories may be applied to find out real-world competitions in business and sports industries where clusters are meaningful. The significant contributions of this study are given in Section 5.1.


### 5.1. The Insights of the Research

(i)
A few properties of semidirected graphs have been established(ii)
Cluster hypergraphs are introduced(iii)
Competition cluster hypergraphs are defined(iv)
m-step competition cluster hypergraphs have been classified(v)
Various features of competition cluster hypergraphs have been studied(vi)
An application/numerical example of COVID-19 networks by using competition cluster hypergraphs has been given


## 5.1. The Insights of the Research

(i)
A few properties of semidirected graphs have been established(ii)
Cluster hypergraphs are introduced(iii)
Competition cluster hypergraphs are defined(iv)
m-step competition cluster hypergraphs have been classified(v)
Various features of competition cluster hypergraphs have been studied(vi)
An application/numerical example of COVID-19 networks by using competition cluster hypergraphs has been given



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