Total and Secure Domination for Corona Product of Two Fuzzy Soft Graphs

Fuzzy sets and soft sets are two different soft computing models for representing vagueness and uncertainty. On the other domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This concept was introduced by [Benecke S, Cockayne EJ, Mynhardt CM. Secure total domination in graphs. Util Math. 2007;74:247–259.] in 2007 and Go and Canoy continue the study of these notions [Canoy RS, Go CE. Domination in the corona and join of graphs. Int Math Forum. 2011;6(16):763–771.] and afterward introduce total dominating and secure total dominating sets. Also graph operations like corona product play a very important role in mathematical chemistry, since some chemically interesting graphs can be obtained from some simpler graphs by different graph operations. In this paper, we characterised the dominating, total dominating, and secure total dominating sets in the corona of two fuzzy soft connected.


Introduction
Molodtsov [1] introduced the concept of soft set that can be seen as a new mathematical theory for dealing with uncertainties. Molodtsov applied this theory to several directions [1][2][3] and then formulated the notions of soft number, soft derivative, soft integral, etc. in Molodtsov et al. [4]. The soft set theory has been applied to many different fields with greatness. Maji [5] worked on theoretical study of soft sets in detail. The algebraic structure of soft set theory dealing with uncertainties has also been studied in more detail. Aktas and Cagman [6] introduced definition of soft groups, and derived their basic properties. The most appreciate theory to deal with uncertainties is the theory of fuzzy sets, developed by Zadeh in 1965. But it has an inherent difficulty to set the membership function in each particular case.
Maji et al. [7] presented the concept of fuzzy soft sets by embedding ideas of fuzzy set in Zadeh [8]. In fact the notion of fuzzy soft set is more generalised than fuzzy set and soft set. Thereafter many papers devoted to fuzzify the concept of soft set theory which leads to a series of mathematical models such as fuzzy soft set [9][10][11][12], generalised fuzzy soft set [1,13], possibility fuzzy soft set [14] and so on. Thereafter Maji and his coauthor [15] introduced the notion of intuitionistic fuzzy soft set which is based on a combination of intuitionistic fuzzy sets and soft set models and they studied the properties of intuitionistic fuzzy soft set.
The first definition of fuzzy graphs was proposed by Kauffman [16] in 1973, from Zadeh's fuzzy relations [8]. But Rosenfeld [17] introduced another elaborated definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of notions of graph theory.
Soft graph was introduced by Thumbakara and George [18]. In 2015, Mohinta and Samanta [19] introduced the concept of fuzzy soft graph and A. Somasundram and S. Somasundram discussed domination in fuzzy graph.
Domination is a rapidly developing area of research in graph theory, and its various applications to networks, distributed computing, social networks and web graphs partly explain the increased interest.
There are other types of domination in graphs which are being studied such as total and secure domination. This concept was introduced by Benecke et al. [20] in 2007 and Go and Canoy continue the study of these notions [21].
In this paper, we characterised the dominating, total dominating and secure total dominating sets in the corona of two fuzzy soft connected graphs.

Preliminaries
First, we review some definitions which can be found in [8,[21][22][23][24][25][26][27][28][29]. By a graph, we mean a pair G * = (V, E), where V is the set and E is a relation on V. The elements of V are vertices of G * and the elements of E are edges of G * . We call V(G * ) the vertex set and E(G * ) the edge set of G * . A fuzzy set A on a set V is characterised by its membership function The underlying crisp graph of a fuzzy graph G = (σ , μ) is denoted by The strength of connectedness between two nodes u,v is defined as the maximum of strengths of all paths between u and v and is denoted by ). A fuzzy graph G = (σ , μ) is said to be a regular if every vertex which is adjacent to vertices having same degrees.
The neighbourhood of v is the set A total dominating set X is a secure total set if for every u ∈ V(G)\X, there exists v ∈ X such that uv ∈ E(G) and [X\{ v}] ∪ { u} is a total dominating set. The domination number γ (G), total domination number γ t (G) or secure total domination γ st (G) of G is the cardinality of a minimum dominating set of G.

Basic Definitions of Fuzzy Soft Graph
Let U be an initial universal set and E be a set of parameters. Let I U denotes the collection of all fuzzy subsets of U and A ⊆ E.
if e ∈ A and 0 denotes the null fuzzy set. The set of all fuzzy soft sets over (U, E) is denoted by FS(U,E).

Definition 3.2: Let
, ∀ e ∈ A and ∀i, j = 1, 2, . . . , n, and this fuzzy soft graph is denoted by G A, V . Definition 3.6: Let G * = (V, E) be a crisp graph and G be a fuzzy soft graph of G * .Then G is said to be a regular fuzzy soft graph if H(e) is a regular fuzzy graph for all e ∈ A. If H(e) is a regular fuzzy graph of degree r for all e ∈ A, then G is a r-regular fuzzy soft graph.
Definition 3.11: Let G 1 and G 2 be two soft graphs of G * 1 and G * 2 , respectively such that A ∩ B = ∅. Their restricted products is defined by G 1 ⊗ G 2 and is defined by Definition 3.12: [26] Let L * be the Cartesian product of two simple graphs G * 1 and G * 2 . Let G 1 and G 2 be, respectively, soft graphs of is a soft graphs of L * Definition 3.13: Let G 1 and G 2 be two soft graphs of G * 1 and G * 2 , respectively such that A ∩ B = ∅. The composition of G 1 and G 2 denoted by G 1 [G 2 ] and is defined by

(v). The minimum fuzzy cardinality of a dominating set in G is called the domination number of G and denoted by γ (G).
A subset T of V(G) is said to be a total dominating set if every vertex in V(G) is dominated by a vertex in T. The minimum fuzzy cardinality of a total dominating set is called the total domination number and denoted by γ t (G). Such a dominating set with minimum fuzzy cardinality is called a minimal dominating set of G.

Proof: Let D be a dominating set in G(e i ) • H(e i ) and let
For the converse, suppose that   Table 1 and is shown in Figure 1 and H 3 , v 4 } and described by Table 2 and is shown in Figure 2.

Proof:
Let v ∈ S and let S * = S\{v}. Suppose S * is not a dominating set of G. Then there exists z ∈ V(G)\S * such that zw / ∈ E(G) for all w ∈ S * .Then z = v and v is the only element of S with zw / ∈ E(G) However, the set D\{v} ∪ {z} cannot be a total dominating set because zw / ∈ E(G) for all w ∈ S * . This contradicts the fact that S is a secure total dominating set of G. Therefore, S\{v} is a dominating set of G. Moreover, if S is a minimum secure total dominating set of G, then the result implies that γ (G) ≤ γ st (G) − σ (v). Suppose the condition holds for D. Then D is clearly a total dominating set of G(e i ) • H(e i ). Let x ∈ V(G(e i ) • H(e i ))\D and v ∈ V(G(e i )) let such that x ∈ V(v + H(e i ) v ). Consider the following cases:

Conclusion
Graph products that allow the mathematical design of a network in terms of small sub graphs that directly express many problems. The result is a flexible algebraic description of networks suitable for manipulation and proof. For graphical research the fuzzy total domination and secure domination are very useful for solving wide range of problems. In this paper we have studied the concepts of fuzzy total domination number and fuzzy secure total domination number for corona product of two fuzzy graphs.