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.
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 be a digraph, which corresponds to a food web. A vertex represents a species in the food web, and an arc means preys on the species . The competition graph of a digraph is an undirected graph which has the same vertex set and has an edge between two distinct vertices if there exists a vertex and arcs .
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 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 holes. Factor and Merz introduced the -step competition graph [10] of a tournament and extended the -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.
Let be a finite set and let be a family of subsets of X such that
The pair is called a hypergraph with vertex set and hyperedge set . The elements of X are vertices of hypergraph H, and the sets 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 be a nonempty set of elements, called vertices or nodes. Also, let where is a set of unordered pairs of vertices, i.e., , called a set of undirected edges, and is a set of ordered pair of vertices, , called a set of directed edges. Here, 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 vertex is denoted as a triplet where is the number of all incident edges of in , is the number of out-directed edges of from the vertex , and is the number of in-directed edges of towards the vertex . Now, the incidence number of a vertex is denoted as and defined in.
Example 1. The degree of a vertex of the graph, as shown in Figure 1, is given as . Now, the incidence number is .
A related property has been developed as follows.
Theorem 1. The sum of the incidence numbers in a semidirected graph is always even.
Proof. Let be a semidirected graph with vertices. If , , , and denote the incidence number, degree, in-degree, and out-degree of a vertex , respectively, then and henceSince one undirected edge contributes two degrees, then . 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 .
Hence, = 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 Figure 2, 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.
Definition 5. Neighbourhood, out-neighbourhood, and in-neighbourhood of a vertex in a semidirected graph are denoted as and defined as follows: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 Figure 3,Thus, is a maximal out-neighbourhood set and is a maximal in-neighbourhood set of the semidirected graph assumed in Figure 3.
Definition 6. The neighbourhood, out-neighbourhood, and in-neighbourhood of a vertex in a semidirected graph are denoted as and defined as follows:
Example 4. In Figure 3, , , and .
Definition 7. The -step semidirected graph of a semidirected graph is defined as follows:
(i) Vertex set of is
(ii) Edge set of is ,
Example 5. In Figure 4, the corresponding 2-step semidirected graph of Figure 3 is shown.
3. Cluster Hypergraphs
Definition 8. Let be a nonempty set and be a subset of such that and . Now, be a multiset whose elements belong to such that(i)(ii)For each element , there exists at least one element such that Then, is said to be cluster hypergraph where is said to be the vertex set and is said to be the multi-hyperedge set.
The k-cluster hypergraph is defined as follows.
Definition 9. Let be a nonempty set and be a subset of , such that and . Now, be a multiset whose elements belong to such that(i)(ii)For each element , there exists at least one element such that Then, is said to be k-cluster hypergraph where is said to be the vertex set and is said to be the multi-hyperedge set. Generally, for , -cluster hypergraphs are assumed as cluster hypergraphs.
Example 6. In Figure 5, , , and It can be easily verified that for each element e in E, there exists an element such that . For example, for there exists an element
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 in is allowed.(iii)In a virtual representation (Figure 6) 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 of 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 is a simple as well as a maximal node, but the node is a maximal node.
3.1.1. Uniform Cluster Hypergraphs
A cluster hypergraph is said to be a-uniform cluster hypergraph if every edge of the hypergraph contains exactly nodes and each maximal node contains simple nodes. In Figure 8, a -uniform cluster hypergraph is shown.
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 Figure 7, a CCCH is shown.
3.1.3. Completeness Property of Cluster Hypergraphs
Let X be a nonempty set containing elements. A cluster hypergraph on contains maximum number of vertices, i.e., , and for the complete cluster hypergraph, the number of edges is . 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.
(2) Complete Uniform Cluster Hypergraph. A complete uniform cluster hypergraph is a -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.
Theorem 2. A complete -uniform cluster hypergraph having cluster nodes contains edges.
Proof. Let us consider a complete -uniform cluster hypergraph having cluster nodes. The hypergraph has cluster nodes containing nodes per cluster. Thus, the total number of simple nodes is . Also, the graph is complete. Therefore, each edge contains exactly nodes. Hence, the total number of edges per cluster is . Also, the total number of edges among clusters is . Thus, the number of edges in the complete -uniform cluster hypergraph having cluster nodes is .
Note. A -uniform complete CCCH having cluster nodes contains and 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. Let be a semidirected graph where is a nonempty vertex set, is the set of undirected edges, and is the set of directed edges. Now, the competition cluster hypergraph of is denoted as where is the vertex set of such that , if and if forms a maximal clique in and which is the hyperedge set if there exists an edge containing vertices if and .
Example 7. Let us consider a semidirected graph, as shown in Figure 11(a). The corresponding competition cluster hypergraph is shown in Figure 11(b). In the semidirected graph, , and also forms a clique. So, the vertex set of the corresponding competition hypergraph is . Now, and , Thus, the edge set of .
(a)
(b)
Proposition 1. Let be a semidirected graph and the corresponding competition cluster hypergraph of be . The number of edges in is equal to the number of a maximal in-degree set of vertices in with cardinality greater than one.
Proof. Let be a semidirected graph where is a nonempty vertex set and the corresponding competition cluster hypergraph of be . In competition cluster hypergraphs, an edge exists between two vertices if they have a common vertex (or vertices). Along with this, if it is found that the third vertex, say , has the same common vertex, then that edge will contain all the three vertices , and and so on. Thus, the number of edges in is equal to the number of a maximal in-degree set of vertices in with cardinality greater than one.
Remark 2. Let be a semidirected graph and the corresponding competition cluster hypergraph of be . The number of cluster nodes in is equal to the number of cliques by undirected edges in .
Theorem 3. Let be a semidirected graph and the corresponding competition cluster hypergraph of be . The number of maximal nodes in is equal to a number of nodes which are not adjacent to other vertices by undirected edges in (the number of undirected edges which are not part of any cliques in ) (number of maximal cliques in ).
Proof. Let be a semidirected graph where is a nonempty vertex set and the corresponding competition cluster hypergraph of be .
Case 1. (): if the graph has no undirected edges, then every node in is a simple node. Hence, the statement is obvious.
Case 2. (): in this case, contains undirected edges. Those edges may construct maximal cliques or simple undirected edges. One maximal clique in will correspond to one cluster node in . 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 is equal to the number of nodes which are not adjacent to other vertices by undirected edges in (the number of undirected edges which are not part of any cliques in ) (number of maximal cliques in ).
Corollary 1. Let be a semidirected graph and the corresponding competition cluster hypergraph of be . All cliques in correspond to maximal nodes if the cliques are disjoint completely, i.e., no vertex and edges are common.
Definition 11. (double competition cluster hypergraphs). Let be a semidirected graph where is a nonempty vertex set, is the set of undirected edges, and is the set of directed edges. Now, the competition cluster hypergraph of is denoted as where is the vertex set of such that , if and if forms a maximal clique in and which is the edge set if there exists an edge containing vertices if and .
Example 8. A semidirected graph is shown in Figure 12(a). The corresponding double competition cluster hypergraph is shown in Figure 12(b).
(a)
(b)
Definition 12. ( competition cluster hypergraphs). Let be a semidirected graph where is a nonempty vertex set, is the set of undirected edges, and is the set of directed edges. Now, the -step competition cluster hypergraph of is denoted as where is the vertex set of such that , if and if forms a maximal clique in and which is the hyperedge set if there exists an edge containing vertices if and .
Example 9. Let us consider a semidirected graph, as shown in Figure 3. The corresponding -step competition cluster hypergraph is shown in Figure 13. In the semidirected graph, , and also . So, the vertex set of the corresponding -step competition hypergraph is . Now, . Thus, the edge set of .
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 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 is an isolated node in the cluster of the cluster hypergraph.
Definition 14. Consider a cluster hypergraph Then, the competition number of is the minimum number of maximal isolated nodes with which forms the competition graph of a semidirected graph.
Example 10. Consider a cluster hypergraph (Figure 16) where and . Then, the corresponding semidirected graph (Figure 17) is drawn. There is no isolated node. Hence, the competition number of is
Algorithm 1. The steps to find the competition number of a cluster hypergraph are given as follows: Step 1. Consider a cluster hypergraph Step 2. Draw the required directed or undirected edges to fit a corresponding semidirected graph from Step 3. The extra nodes if needed to get may be taken as isolated nodes Step 4. The minimum number of maximal isolated nodes will be the competition number of
Proposition 2. Competition number of a cluster hypergraph is zero.
Proof. Consider a 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 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 .
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 . Then, there exist edges , ; , ;…, ; for some integer . Now, = . Since an edge implies there exists a path of length from to and similar for ; hence . Similarly, if an edge then it implies . Therefore, .
Proposition 3. In a semidirected graph , if , then is a null cluster hypergraph.
Proof. Since is an -step competition cluster hypergraph of and if , then there does not exist a path of length in Hence, for all . Thus, there does not exist any edge in . Therefore, 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 Tables 1 and 2, the corresponding semidirected graph is shown in Figure 12, and its corresponding competition cluster hypergraph is shown in Figure 13.
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 emissions for a small semidirected graph. All the assumed countries are affected by COVID-19 and emissions. Thus, there will be direct edges from COVID-19 and 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 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 like m-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
Data Availability
The data, provided in the article, were collected from Wikipedia and https://www.worldometers.info/coronavirus/#countries. They are available in the public domain, so the authors have no restriction on that.
Conflicts of Interest
The authors declare that they have no conflicts of interest.