Some Characterizations of Whole Edge Domination in Bipolar Fuzzy Graphs

For a connected graph $$G = (V, E)$$ , a subset F of E is an edge dominating set (resp. a total edge dominating set) if every edge in $$E-F$$ (resp. in E) is adjacent to at least one edge in F, the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of G is the edge domination number (resp. total edge domination number) of G, denoted by $$\gamma '(G)$$ (resp. $$\gamma '_t(G)$$ ). In the present paper, we study a parameter, called the semitotal edge domination number, which is squeezed between $$\gamma '(G)$$ and $$\gamma '_t(G)$$ . A semitotal edge dominating set is an edge dominating set S such that, for every edge e in S, there exists such an edge $$e'$$ in S that e either is adjacent to $$e'$$ or shares a common neighbor edge with $$e'$$ . The semitotal edge domination number, denoted by $$\gamma ^{'}_{st}(G)$$ , is the minimum cardinality of a semitotal edge dominating set of G. In this paper, we prove that the problem of deciding whether $$\gamma ^{'}(G)=\gamma ^{'}_{st}(G)$$ or $$\gamma _t^{'}(G)=\gamma ^{'}(G)$$ is NP-hard even when restricted to planar graphs with maximum degree 4. We also characterize trees with equal edge domination and semitotal edge domination numbers (Pan et al. in The complexity of total edge domination and some related results on trees, J Comb Optim, 2020, https://doi.org/10.1007/s10878-020-00596-y , we characterized trees with equal edge domination and total edge domination numbers).

Bipolar fuzzy graph is more precise than a fuzzy graph when dealing with imprecision as it is focusing on the positive and negative information of each vertex and edge. Nowadays, researchers have utilized bipolar fuzzy graphs in decision-making problems. Bipolar fuzzy competition graphs aid to compute the competition between the vertices in bipolar fuzzy graphs. To depict the best competitions among the competitions of bipolar fuzzy graphs, the best bipolar fuzzy competition graph can be defined using bipolar fuzzy α-cut and the strength of the competition between the vertices can also be determined. Fuzzy graphs are used well to frame modelling in real-time problems. In particular, when the real-time scenario is modelled using the bipolar fuzzy graph, it gives more precision and flexibility. At present, researchers have focused on decision-making techniques with bipolar fuzzy graphs. The DEMATEL method is one of the powerful decision-making tools. It effectively analyses the complicated digraphs and matrices. The fuzzy DEMATEL technique can convert the interrelations between factors into an intelligible structural model of the system and divide them into cause and effect groups. Therefore, this study attempts to design the DEMATEL method under the bipolar fuzzy environment. To illustrate this proposed technique, the problem of identifying the best mobile network is taken. With this method, the benefits and drawbacks of networks are measured and a complicated bipolar fuzzy directed graph can be transformed into a viewed structure.

Applications using domination in graphs can be found across multiple domains.When there is a set number of resources (such as fire departments and healthcare facilities) and the goal is to reduce the distance that someone must travel in order to reach the most nearby facility, domination emerges in facility positioning problems.Domination notions can also be found in land mapping problems (e.g., limiting the quantity of places where an assessor has to visit in order to obtain measurements of elevation for an entire region), tracking telecommunications or electrical infrastructure, and tasks involving spotting squads of senators. A comparable issue arises when efforts are made to minimize the quantity of facilities needed to serve every individual and the ideal distance to service is established. Considering the graph G = [ {V}, {E}]. Let the set I V {G} is a secure - vertex - edge dominating set of G, suppose every edge, y E [G], then there exists a vertex V I so that V stands up for y . i.e., The vertex in I defends the edges incident on that vertex and the edges which lie next to the incident edges. A secure - vertex - edge dominating set I of a graph G has the characteristic of being a dominant set where every vertex z V – I either follows a vertex or a vertex adjacent to the incident edges of z, x I such that (I- {x}) {z} is a dominating set. The secure - vertex - edge domination number in G is the least cardinality of secure - vertex - edge domination and is depicted by . We have commenced researching this new parameter and have found the secure - vertex - edge dominance number of several standard graphs and the middle graphs of some standard graphs. In the current analysis, the secure - vertex - edge dominance number of a few designated specific graphs such as Bull Graph, Durer Graph, Heawood Graph, Moser Spindle Graph and etc., was discovered.

Topological indices are the numbers that remain constant under graph automorphism. Topological indices describe a network’s connectivity, structure, and topological characteristics. These indices have many applications in crisp graphs. However, in many cases, it is observed that some situations can’t be described using the idea of crisp graphs. So, to overcome this issue, the need to define topological indices for fuzzy and bipolar fuzzy graphs arises. The F-index, or the Forgotten Index, is a significant topological index. A bipolar fuzzy graph with two opposite-sided opinions of both the edges and vertices measures the impreciseness or uncertainties of the edges and vertices along the positive and negative sides. In this article, we have presented the Forgotten Index for bipolar fuzzy graphs. Then, we have proved some theorems regarding the F-index of numerous types of bipolar fuzzy graphs, such as regular bipolar fuzzy graphs, complete bipolar fuzzy graphs, etc., the bounds of the F-index in bipolar fuzzy graphs, and the relationships of the F-index with other topological indices in bipolar fuzzy graphs. We have applied the proposed topological index, the F-index for bipolar fuzzy graphs, to matrimonial websites to find potential life partners based on compatibility and discussed the application of the Forgotten Index in gene regulatory networks.

The main purpose of this paper is to introduce the notion of bipolar f -morphism on bipolar fuzzy graphs and regular bipolar fuzzy graphs. We study the action of bipolar f -morphism on bipolar fuzzy graphs and derive some elegant results on weak and co-weak isomorphism. Also, we define d 2 -degree and total d 2 -degree of a vertex in bipolar fuzzy graphs and study (2, k )-regularity and totally (2, k )-regularity of bipolar fuzzy graphs. d 2 -degree and total d 2 -degree of a vertex in bipolar fuzzy graphs are highly utilized by computer science, geometry, algebra, number theory and operation research. In addition these properties will also be helpful to study large bipolar fuzzy graph as a combination of small, bipolar fuzzy graphs and to derive its properties from those of the smaller ones.

In this chapter, we discuss the concept of irregularity in bipolar fuzzy graphs and present isomorphism properties of regular, m-totally regular, neighborly irregular, totally irregular, highly irregular, and neighborly totally irregular bipolar fuzzy graphs. We present certain characterizations under which, regular and totally regular bipolar fuzzy graphs, and highly irregular and neighborly irregular bipolar fuzzy graphs are equivalent. We discuss certain formulae of order and size of \(m-\)totally regular bipolar fuzzy graphs. We study the concept of bipolar fuzzy line graphs, and establish a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph.

In this chapter, we first review the notion of bipolar fuzzy sets and present several basic concepts concerning bipolar fuzzy graphs and bipolar fuzzy digraphs. We discuss different methods of construction of bipolar fuzzy graphs and their isomorphism properties. We describe certain types of bipolar fuzzy graphs, bipolar fuzzy walk, bipolar fuzzy bridge, strength of connectedness, weak and strong bipolar fuzzy edges. We establish the relations on bipolar fuzzy graphs, complement of bipolar fuzzy graphs, and crisp graphs with different operations, \(\alpha -\)cuts and \((\alpha ,\beta )-\)cuts. We also study certain operations and properties of complex bipolar fuzzy graphs. Moreover, with the help of composition of bipolar fuzzy relations, connectivity, and weighted matrices, we study the importance of bipolar fuzzy digraphs with a number of real-world problems. This chapter is basically adapted from [1, 2, 3, 45, 46, 51].

In this paper we discussed the prominence of Bipolar fuzzy graphs (BFG). Fuzzy set assigns a sequence of membership values to the elements of the universal set ranging from 0 to 1, whereas now our study about Bipolar fuzzy graphs whose membership degree range is [-1, 1]. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. When we assess the position of an object in space, we may have positive information expressed as a set of possible places and negative information expressed as a set of impossible places. This corresponds to the idea that the union of positive and negative information does not cover the whole space. Dominating sets have a vital function regarding the theory of fuzzy graphs. Traveling salesman problem, communication network, traffic route problem are largely discussed applications among the diverse applications dealing with the theory of dominations. In this paper we generalized Bipolar fuzzy graphs and explored various types of dominations on Bipolar fuzzy graphs such as Strong Dominations, Split and non-split dominations, Multiple dominations and some applications of Bipolar fuzzy graphs. Fuzzy graphs found an increasing number of applications in modeling real time systems where the information inherent in the system varies with different levels of precision. Bipolar fuzzy graphs can be used to model many problems in economics, operations research, etc; involving two similar, but opposite type of qualitative variables like success and failure, gain and loss etc.

Bipolar fuzzy graphs, capable of capturing situations with both positive and negative memberships, have found diverse applications in various disciplines, including decision-making, computer science, and social network analysis. This study investigates the domain of domination and global domination numbers within bipolar fuzzy graphs, owing to their relevance in these aforementioned practical fields. In this study, we introduce certain operations on bipolar fuzzy graphs, such as intersection, join, and union of two graphs. Furthermore, we analyze the domination number and the global domination number for various operations on bipolar fuzzy graphs, including intersection, join, and union of fuzzy graphs and their complements.

Theoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. In this paper, we discussed some properties of the µ–complement of bipolar fuzzy graphs. Self µ–complement bipolar fuzzy graphs and self weak µ–complement bipolar fuzzy graphs are defined and a necessary condition for a bipolar fuzzy graph to be self µ–complement is given. We defined busy vertices and free vertices in bipolar fuzzy graphs and studied their image under an isomorphism. Categorical properties of bipolar fuzzy graphs are discussed. Also, we investigated some properties of isomorphism on bipolar fuzzy graphs.

Domination of vertex-edges in bipolar fuzzy graphs

A bipolar fuzzy graph is a generalization of graph theory by using bipolar fuzzy sets. The bipolar fuzzy sets are an extension of fuzzy sets. This paper introduces an effective degree of a vertex, a (ordinary) degree of a vertex in bipolar fuzzy graph as analogous of fuzzy graph, a semiregular bipolar fuzzy graph, and a semicomplete bipolar fuzzy graph. Further, this paper gives some propositions.

A set $$S $$ (of vertices) of a graph $$G$$ is termed a dominating set of $$G$$ if each vertex in $$V - S$$ is adjacent to a node in $$ S$$. A dominating set $$ S$$ such as the subgraph induced by $$S$$ has an isolated vertex is termed an isolate dominating set and also the minimum count of an isolate dominating set is termed the isolate domination number of $$G$$ and it is represented by $$\gamma_{is} (G)$$. A subset $$X \subseteq E $$ is said to be an edge dominating set if each edge in $$X - E $$ is adjacent to some edge in $$S$$. The edge domination number is that the count of the smallest edge dominating set of $$G$$ and is employed by $$\gamma^{\prime}$$. A collection of edges $$X$$ of $$E$$ is claimed to be a perfect edge dominating set if each edge not in $$ X$$ is adjacent to precisely one edge in $$X$$. The ideal edge domination number is that the minimum cardinality has taken perfect edge dominating sets of $$G$$ and is denoted by $$\gamma_{p}^{^{\prime}}$$. In this paper, we initiate the survey of bondage related to isolate domination. The isolate bondage number $$b_{is} (G)$$ is outlined to be the minimum cardinality of a collection of edges whose relieved from $$G$$ ends up in a graph $$G^{\prime}$$ fulfilling $$\gamma_{is} (G^{\prime}) > \gamma_{is} (G)$$. We obtain several results for isolate dominating set and identical values of isolate bondage number. Moreover, we investigate some bounds for the isolate bondage number, and this bound is keen and analyze under which conditions the domination parameter and isolate domination parameter are equal. Also, we found some more results for perfect edge domination, and we characterize trees for which $$\gamma^{\prime} = \gamma_{p}^{^{\prime}}$$ and further exciting results.

A subset $X$ of edges of a graph $G$ is called an \textit{edge dominating set} of $G$ if every edge not in $X$ is adjacent to some edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$ The weight of an edge Roman dominating function $f$ is the value $w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number.

In this chapter, we discuss the notion of distance in bipolar fuzzy graphs and present certain properties concerning distance functions in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs.