An Extension of Fuzzy Competition Graph and Its Uses in Manufacturing Industries

A Pythagorean fuzzy set is very effective mathematical framework to represent parameter-wise imprecision which is the property of linguistic communication. A Pythagorean fuzzy soft graph is more potent than the intuitionistic fuzzy soft as well as the fuzzy soft graph as it depicts the interactions among the objects of a system using Pythagorean membership grades with respect to different parameters. This article addresses the content of competition graphs as well as economic competition graphs like k-competition graphs, m-step competition graphs and p-competition graphs in Pythagorean fuzzy soft environment. All these concepts are illustrated with examples and fascinating results. Furthermore, an application which describes the competition among distinct forest trees, that grow together in the mixed conifer forests of California, for plant resources is elaborated graphically. An algorithm is also designed for the construction of Pythagorean fuzzy soft competition graphs. It is worthwhile to express the competing and non-competing interactions in various networks with the help of Pythagorean fuzzy soft competition graphs wherein a variation in competition relative to different attributes is visible.

Presently, fuzzy graph theory has wide applications in engineering and social networks. Based on the applications, different types of fuzzy graphs are defined. Among them the competition graph is significantly used to model ecological problems. This graph is one kind of intersection graph. Different competitions of real world have been designed by fuzzy competition graphs. One generalization of fuzzy competition graph, called $$m$$ -step fuzzy competition graph, is defined in this paper. Some related fuzzy graphs including fuzzy $$m$$ -step neighbourhood graph, fuzzy economic competition graphs and fuzzy $$m$$ -step economic competition graphs are introduced. Some properties of these new graphs have been investigated.

This study extends the concept of competition graphs to cubic fuzzy competition graphs by introducing additional variations including cubic fuzzy out-neighbourhoods, cubic fuzzy in-neighbourhoods, open neighbourhood cubic fuzzy graphs, closed neighbourhood cubic fuzzy graphs, cubic fuzzy (k) neighbourhood graphs and cubic fuzzy [k]-neighbourhood graphs. These variations provide further insights into the relationships and competition within the graph structure, each with its own defined characteristics and examples. These cubic fuzzy CMGs are further classified as cubic fuzzy k-competition graphs that show competition in the k th order between components, p -competition cubic fuzzy graphs that concentrate on competition in terms of membership degrees, and m -step cubic fuzzy competition graphs that analyze competition in terms of steps. Further, some related results about independent strong vertices and edges have been obtained for these cubic fuzzy competition graph classes. Finally, the proposed concept of cubic fuzzy competition graphs is supported by a numerical example. This example showcases how the framework of cubic fuzzy competition graphs can be practically applied to the predator-prey model to illustrate the representation and analysis of ambiguous information within the graph structures.

In 1968, Cohen [4] first introduced competition graph while working on a problem in ecology. Competition graph (CompG) is one kind of intersection graph and it is originally used for modeling ecological problems. Though this graph can be used to model various types of problems in the real world, Cohen used the crisp competition graph to model the ecological problem and hence the vertices and edges of such graphs were precisely defined.

Fuzzy competition graph as a generalization of competition graph is introduced here. Two generalizations of fuzzy competition graph as fuzzy k-competition graphs and p-competition fuzzy graphs are also defined. These graphs are related to fuzzy digraphs. Fuzzy neighbourhood graphs, related to fuzzy graphs, are also defined here. Besides, some relations between fuzzy competition graphs and fuzzy neighbourhood graphs have been established. And finally, several results to find strong edges of the above mentioned graphs have also been established.

Fuzzy competition graph as the generalization of competition graph is introduced here. Fuzzy k-competition graph as a special type of fuzzy competition graph is defined here along with fuzzy isolated vertices. The fuzzy competition number is also introduced and investigated several properties. Isomorphism properties on fuzzy competition graphs are discussed.

The competition-common enemy graph of a digraph

An [Formula: see text]-polar fuzzy model is a generalization of the fuzzy model. This model is useful for multi-polar information, multi-agent, multi-attribute and multi-object network models. In this research paper, we first introduce certain new concepts, including [Formula: see text]-polar fuzzy digraphs, [Formula: see text]-polar fuzzy-out neighborhood, [Formula: see text]-polar fuzzy-in neighborhood, [Formula: see text]-polar fuzzy competition graphs and [Formula: see text]-polar fuzzy [Formula: see text]-competition graphs. Moreover, we describe the construction of [Formula: see text]-polar fuzzy competition graphs of certain products of [Formula: see text]-polar fuzzy digraphs from their respective [Formula: see text]-polar fuzzy competition graphs. We present some applications of [Formula: see text]-polar fuzzy competition graphs in food webs. Finally, we present certain algorithms for computing strength of competition between species.

A complex fuzzy set (CFS) is a remarkable generalization of the fuzzy set in which membership function is restricted to take the values from the unit circle in the complex plane. A CFS is an efficient model to deal with uncertainties of human judgement in more comprehensive and logical way due to the presence of phase term. In this research article, we introduce the concept of competition graphs under complex fuzzy environment. Further, we present complex fuzzy k-competition graphs and p-competition complex fuzzy graphs. Moreover, we consider m-step complex fuzzy competition graphs, complex fuzzy neighborhood graphs (CFNGs), complex fuzzy economic competition graphs (CFECGs) and m-step complex fuzzy economic competition graphs with interesting properties. In addition, we describe an application in ecosystem of our proposed model. We also provide comparison of proposed competition graphs with existing graphs.

A complex intuitionistic fuzzy set (CIFS) has an ability to represent the problems with intuitionistic uncertainty and periodicity, simultaneously. In this paper, we present a new framework for handling complex intuitionistic fuzzy information by combining the CIFSs with competition graphs. We first introduce the concept of complex intuitionistic fuzzy competition graphs along with its two worthwhile extensions, namely, complex intuitionistic fuzzy k-competition and complex intuitionistic fuzzy p-competition graphs. Further, we present complex intuitionistic fuzzy neighborhood graphs and m-step complex intuitionistic fuzzy competition graphs with some of their remarkable results. We also describe an application of complex intuitionistic fuzzy competition graphs along with an algorithm which finds the competition among the applicants competing for the particular jobs possessing the two-dimensional information. Moreover, we provide a comparison analysis of proposed competition graphs with existing graphs to find the validity and superiority of proposed competition graphs.

Connected triangle-free [formula omitted]-step competition graphs

The m-step, same-step, and any-step competition graphs

The notion of competition graphs has a wide range of applications and has been utilized in designing various ecological problems and real-world competitions. Here, the idea of m-step inverse fuzzy mixed competition graphs has been introduced and some of its basic properties have been discussed. Also, some ideas associated with the concept of m-step inverse fuzzy mixed competition graphs, such as m-step inverse fuzzy neighborhood graph, inverse fuzzy mixed economic competition graph, and m-step inverse fuzzy mixed economic competition graph have been defined. Our motive was to focus on the real-life problems that can be properly addressed with graphs involving directed and undirected edges. In this case, the concept of an m-step fuzzy competition graph is not appropriate for solving these problems. However, the idea of m-step inverse fuzzy mixed competition graphs is more relevant to address them. An application of m-step inverse fuzzy mixed competition graphs has been identified as a suitable solution to a real-life problem.

In this paper, the notion of picture fuzzy competition graph along with its few generalizations such as m-step picture fuzzy competition graphs, picture fuzzy economic competition graphs and picture fuzzy competition hypergraphs are introduced. Some related picture fuzzy graphs including picture fuzzy m-step neighborhood graph, picture fuzzy m-step economic competition graph and picture fuzzy k-competition hypergraphs are introduced. Some properties of these graphs have been investigated. Finally, applications of m-step picture fuzzy competition graphs and picture fuzzy competition hypergraphs are presented in several fields such as in education system, ecosystem, business market and job competition.

The competition graph and its generalizations have interested graph theorists for over 30 years. Recently, Cho, Kim, and Nam introduced the m-step competition graph and computed the 2-step competition numbers of paths and cycles. We extend their results in a partial determination of the m-step competition numbers of paths and cycles. In addition, we introduce two new variants of competition graphs: same-step and any-step. We classify same-step and any-step competition graphs and investigate their related competition numbers.