Domination in Fuzzy Directed Graphs

: A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let G ∗ D = ( V , A ) be a directed simple graph, where V is a ﬁnite nonempty set and A = { ( x , y ) : x , y ∈ V , x (cid:54) = y } . A fuzzy digraph G D = ( σ D , µ D ) is a pair of two functions σ D : V → [ 0,1 ] and µ D : A → [ 0,1 ] , such that µ D (( x , y )) ≤ σ D ( x ) ∧ σ D ( y ) , where x , y ∈ V . An edge µ D (( x , y )) of a fuzzy digraph is called an effective edge if µ D (( x , y )) = σ D ( x ) ∧ σ D ( y ) . Let x , y ∈ V . The vertex σ D ( x ) dominates σ D ( y ) in G D if µ D (( x , y )) is an effective edge. Let S ⊆ V , u ∈ V \ S , and v ∈ S . A subset σ D ( S ) ⊆ σ D is a dominating set of G D if, for every σ D ( u ) ∈ σ D \ σ D ( S ) , there exists σ D ( v ) ∈ σ D ( S ) , such that σ D ( v ) dominates σ D ( u ) . The minimum dominating set of a fuzzy digraph G D is called the domination number of a fuzzy digraph and is denoted by γ ( G D ) . In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.


Introduction
Within the domains of graph theory, a directed graph is an ordered triple (V(D), A(D), ψ D ) consisting of a nonempty set V(D) of vertices; a set A(D), disjointed from V(D), of arcs; and an incidence function ψ D that associates with each arc of D an ordered pair of vertices of D [1]. If a is an arc and u and v are vertices such that ψ D (a) = (u, v), then a is said to join u to v; u is the tail of a and v is its head. For convenience, a directed graph is abbreviated to digraph. For a comprehensive discussion of graph theory, we refer to [2]. On the other hand, the concept of a fuzzy set was introduced in a seminal paper presented in 1965 by Zadeh [3]. Rosenfeld [4] explored the fuzzy relations on fuzzy sets and introduced fuzzy graphs in 1975. Some fundamental operations of fuzzy graphs were introduced by Mordeson and Chang-Shyh [5], and the latest collection of some important developments on the theory and applications of fuzzy graphs was compiled by Mordeson and Nair [6]. Since then, various extensions of fuzzy graphs were offered in the literature, including M-strong fuzzy graphs [7], intuitionistic fuzzy graphs [8], regular fuzzy graphs [9], bipolar fuzzy graphs [10], interval-valued fuzzy graphs [11], and Dombi fuzzy graphs [12], among others. Note that this list is not intended to be comprehensive. We review some basic notions of fuzzy graphs by letting S be a set. A fuzzy subset of S is a mapping σ : S → [0, 1] which assigns to each element x ∈ S a degree of membership, 0 ≤ σ(x) ≤ 1. Similarly, a fuzzy relation on S is a fuzzy subset of S × S, that is, a mapping µ : S × S → [0, 1] , which assigns to each ordered pair of elements (x, y) a degree of membership, 0 ≤ µ(x, y) ≤ 1. In a special case where σ and µ can only take on the values 0 and 1, they become the characteristic functions of an ordinary subset of S and an ordinary relation on S, respectively.
With interesting results and an array of applications, domination in a graph has been a vast area of research in graph theory. It was introduced by Claude Berge in 1958 and Oystein Ore in 1962 [13], with the earliest results and applications put forward by Cockayne and Hedetniemi [14]. The most comprehensive reference on the topic can be found in Haynes et al. [15], with more advanced and latest concepts in Haynes [16] and Haynes et al. [17]. Extended forms of domination in graphs have been vast in the domain literature. Some very recent forms include broadcast domination [18], pitchfork domination [19], Roman domination [20], double Roman domination [21], triple Roman domination [22], captive domination [23], outer-convex domination [24], and paired domination [25], among others. The trajectory of these topics has been exponential in the last decade. Consider G = (V(G), E(G)) as a graph. A subset S of a vertex set V(G) is a dominating set of a graph G if, for every vertex v ∈ V(G)\S, there exists a vertex x ∈ S such that xv is an edge of G. The domination number γ(G) of G is the smallest cardinality of a dominating set S of G. As an extension, the concept of domination in fuzzy graphs was introduced by Somasundaram [26]. Let V be a finite nonempty set, and E be a collection of all two-element subsets of V. A fuzzy graph G = (σ, µ) is a set with two functions σ : The minimum fuzzy cardinality of a dominating set in G is called the domination number of G and is denoted by γ(G).
The notion of fuzzy digraphs can be traced back to the work of Mordeson and Nair [27], with recent advances reported by Kumar and Lavanya [28]. A fuzzy di- is a fuzzy relation on V, and µ D is a set of fuzzy directed edges called fuzzy arcs. An indegree of a vertex u in a fuzzy digraph is the sum of the µ D values of the edges that are incident towards the vertex σ D (u). The outdegree of any vertex u in the fuzzy digraph is the sum of membership function values of all those arcs that are incident out of the vertex u. The indegree is denoted by d − (u) and the outdegree by d + (u), where u is any vertex in V. A subset S ⊆ V is a fuzzy out dominating set of G D if, for every vertex v ∈ V − S, there exists The domination in fuzzy digraphs is a new concept in the domain literature, with limited insights. With such a new concept, we propose a new domination parameter in a fuzzy digraph. Motivated by the concepts of fuzzy digraphs [27,28] and the notions of domination of graphs [13], this work intends to advance the literature of domination in a fuzzy graph and a directed graph. All graphs considered in this paper are finite and directed without a loop. We use G * D = (V, A) as a latent directed graph of a fuzzy digraph G D = (σ D , µ D ), where V is a vertex set and A is an arc set of a directed graph G * D , while σ D is a vertex set and µ D is an arc set of a fuzzy digraph G D . A set of vertices S ⊆ V is a dominating set of G * D if each vertex v ∈ V\S is dominated by at least a vertex in S. The domination number γ(G * D ) of G * D is the smallest cardinality of a dominating set S of G * D . In this paper, the concept of domination in a fuzzy digraph is introduced/defined, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled. The contribution of this work lies in providing general results (i.e., theorems, corollaries) of the minimum dominating set of a fuzzy directed graph in order to facilitate new advances on these concepts.

Preliminaries
This section provides a new definition of a fuzzy directed graph, introduces some working terminologies, and gives some useful observations in the form of remarks and examples.
The term digraph is used to represent a directed graph.
V is a set of vertices or nodes of a latent digraph, that is, A is a set of directed edges or arcs of a latent digraph, that is, σ D is a set of vertices or nodes of a fuzzy digraph, that is, µ D is a set of edges or arcs of a fuzzy digraph, that is, Figure 1. Figure 2.
Example 3. Let G D be a directed graph as shown in Figure 3. Then, G D is not a fuzzy digraph because 0.7 0.8 ∧ 0.6. Moreover, 0.9 0.8 ∧ 0.9.    Figure 4, the orderp of G D is Definition 3. An arc µ D ((x, y)) of a fuzzy digraph is called an effective arc if Figure 4, µ D ((a, c)) = 0.8 is the only effective arc of G D .

Domination in Fuzzy Digraphs
In this section, we define a dominating set in a fuzzy digraph G D . Further, we characterize the minimal dominating set of a fuzzy digraph and give some useful results. Then The fuzzy cardinality of a minimum dominating set is called the domination number of G D and is denoted by γ(G D ), that is, where S is a dominating set of G * D .
Example 9. Let G * D and G * D be latent digraphs of fuzzy digraphs G D = (σ D , µ D ) and G D = (σ D , µ D ), respectively (see Figure 6). If  Figure 7).  From the definitions and observations, the following remark is immediate.
The following result gives a characterization of the minimal dominating set of a fuzzy directed graph.
For the converse, the proof is immediate.

Some Special Fuzzy Digraphs
In this section, we introduce the definition of some special fuzzy digraphs G D . Further, we give the general formula of computing the domination number of G D .

Definition 6.
A fuzzy dipath (directed path) P σ D is a sequence of effective arcs having the property that the ending vertex of each arc is the same as the starting vertex of the next arc in the sequence.
The vertices σ D (x 1 ) and σ D (x n ) are the first and last vertex, respectively, of a nontrivial fuzzy dipath.
The following result illustrates the domination number of a fuzzy dipath.

Definition 7.
A fuzzy dicycle (directed cycle) C σ D is a dipath where it starts and ends with the same vertex.
The following result provides the domination number of a fuzzy dicycle.
Theorem 3. Let C σ D = (σ D , µ D ) be a fuzzy dicycle of a latent directed cycle C n = (V, A) where n ≥ 3. Then, one of the following is satisfied: Proof. Let I = {1, 2, . . . , n}. Because C σ D is a dipath that starts and ends with the same node, the arcs µ D (x i , x i+1 ) for all i ∈ I and µ D (x n , x 1 ) are effective. This means that σ D (x i ) dominate σ D (x i+1 ) for all i ∈ {1, 2, . . . , n − 1} and σ D (x n ) dominate σ D (x 1 ). Case 1. If n is an even integer, then n = 2k for some positive integer k. Now, the sets ) . This proves the statement (i). Case 2. If n is an odd integer, then n = 2k − 1 for some positive integer k. Now, the sets Let (j + 1)/2 + i = (n + 1)/2. Then, are other minimal dominating sets of C σ D . Generally, Let j/2 + i = (n + 1)/2. Then, Hence, the domination number of C σ D is γ(C σ D ) = min(X ∪ Y). This proves statement ii) Figure 10).

Conclusions
In this work, we introduced the concept of domination in a fuzzy digraph, provided the characteristics of the minimum dominating set of fuzzy digraphs, and modeled the domination number of a fuzzy dipath and a fuzzy dicycle. The domination number in a fuzzy dipath and a fuzzy dicycle was presented and proved. The immediate consequences of the mentioned concepts were all proved. Some related problems are still open for future work.
(a) Characterize the dominating sets of each of the following special fuzzy digraphs-the wheel W n , the complete bipartite K m,n , the star S n , and the fan F n . (b) Find the domination number of each of the following special fuzzy digraphs: W n , K m,n , S n , and F n .
Aside from these problems, future works could explore the application of domination in fuzzy digraphs in problem structuring methods commonly used in the literature, such as fuzzy decision-making trial and evaluation laboratory (DEMATEL) [29], fuzzy cognitive mapping (FCM) [30], and fuzzy interpretive structural modelling (ISM) [31], among others.