Vertex Domination in t-Norm Fuzzy Graphs

For the first time, We do fuzzification the concept of domination in crisp graph on a generalization of fuzzy graph by using membership values of vertices, α-strong edges and edges. In this paper, we introduce the first variation on the domination theme which we call vertex domination. We determine the vertex domination number γv for several classes of t-norm fuzzy graphs which include complete t-norm fuzzy graph, complete bipartite t-norm fuzzy graph, star t-norm fuzzy graph and empty t-norm fuzzy graph. The relationship between effective edges and α-strong edges is obtained. Finally, we discuss about vertex dominating set of a fuzzy tree with respect to a t-norm ⊗ by using the bridges and α-strong edges equivalence.

The rest of this paper is organized as follows.In Section 2, we lay down the preliminary results which recall some basic concept.In Section 3, the vertex domination number of a t-norm fuzzy graph is defined in a classic way, Definition (3.2), (3.4), (3.5).

Preliminary
We provide some basic background for the paper in this section.Definition 2.1 (Ref.[17], Definition 5.1.1,pp.82, 83).A binary operation We concern with a t-norm fuzzy graph which is defined on a crisp graph.So we recall the basic concepts of crisp graph.
A graph (Ref.[1], p. 1) G is a finite nonempty set of objects called vertices (the singular is vertex ) together with a (possibly empty) set of unordered pairs of distinct vertices of G called edges.The vertex set of G is denoted by V (G), while the edge set is denoted by E(G).
We recall that a fuzzy subset in Ref. we lay down the preliminary results which recall some basic concepts of fuzzy graph from Ref. [11].
A fuzzy graph in Ref. ( [11], p. 19) is denoted by G = (V, σ, µ) such that µ({x, y}) ≤ σ(x) ∧ σ(y) for all x, y ∈ V where V is a vertex set, σ is a fuzzy subset of V , µ is a fuzzy relation on V and ∧ denote the minimum.We call σ the fuzzy vertex set of G and µ the fuzzy edge set of G, respectively.We consider fuzzy graph G with no loops and assume that V is finite and nonempty, µ is reflexive (i.e., µ({x, x}) = σ(x), for all x) and symmetric (i.e., µ({x, y}) = µ({y, x}), for all x, y ∈ V ).In all the examples σ and µ is chosen suitably.In any fuzzy graph, the underlying crisp graph is denoted by G * = (V, E) where V and E are domain of σ and µ, respectively.The fuzzy graph Similarly, the fuzzy graph H = (τ, ν) is called a fuzzy subgraph of G = (V, σ, µ) induced by P in if P ⊆ V, τ (x) = σ(x) for all x ∈ P and ν({x, y}) = µ({x, y}) for all x, y ∈ P. For the sake of simplicity, we sometimes call H a fuzzy subgraph of G.We say that the partial fuzzy subgraph (τ, ν) spans the fuzzy graph (σ, µ) if σ = τ.In this case, we call (τ, ν) a spanning fuzzy subgraph of (σ, µ).
For the sake of simplicity, we sometimes write xy instead of {x, y} fuzzy cycle, if it contains more than one weakest edge.The strength of a cycle is the strength of the weakest edge in it.The strength of connectedness between two vertices x and y in is defined as the maximum of the strengths of all paths between x and y and is denoted by µ ∞ G (x, y).
Definition 2.2 (Ref.[10], Definition 3.1, p. 131).Let G = (V, E) be a graph.Let σ be a fuzzy subset of V and µ be a fuzzy subset of E.
Let k be a positive integer.Define

Main Results
In this section, we provide the main results.
Let x, y ∈ V. We say that x dominates y in G as α-strong if the edge {x, y} is α-strong.
It is well known and generally accepted that the problem of determining the domination number of an arbitrary graph is a difficult one.Because of this, researchers have turned their attention to the study of classes of graphs for which the domination problem can be solved in polynomial time.
Proof.Since G = (σ, µ) be a complete t-norm fuzzy graph with respect to ⊗, none of edges are α-strong by Corollary (3.9).so we have    Proof.Let G = (σ, µ) be a complete bipartite t-norm fuzzy graph with respect to a that induce uv is not α-strong edge.The result follows.
Proof.Obviously, the result is hold by using Proposition (3.14).
Proposition 3.16 (Star t-norm fuzzy graph).Let G = (σ, µ) be a star t-norm fuzzy graph with respect to a t-norm ⊗.
Proof.Let G = (σ, µ) be a star t-norm fuzzy graph with respect to a t-norm ⊗.Let Proposition 3.17 (Complete bipartite t-norm fuzzy graph).Let G = (σ, µ) be a star t-norm fuzzy graph with respect to a t-norm ⊗ which is not star t-norm fuzzy graph.
Proof.Let G = K 1,σ be complete bipartite t-norm fuzzy graph with respect to ⊗.Then both of V 1 and V 2 include more than one vertex.In K σ1,σ2 , none of edges are α-strong by Proposition (3.14).Also, each vertex in V 1 is adjacent with all vertices in V 2 and conversely.Hence in K σ1,σ2 , the α-strong dominating sets are V 1 and V 2 and any sets containing 2 vertices, one in V 1 and other in V 2 .Hence . So the proposition is proved.(1) xy is a bridge with respect to ⊗; (2) µ ∞ ⊗ (x, y) < µ(xy); (3) xy is not a weakest edge of any cycle.Proof.Let uv be a edge of (σ, µ).So µ(uv) = µ ∞ ⊗ (u, v) by Proposition (3.22).Hence µ(uv) ≤ µ ∞ ⊗ (u, v).It means the edge uv is not α-strong.
The following example illustrates this concept.

Applications
According to some applications of t-norm fuzzy graph in Ref. ( [10], Section 4, p.137), increasing numbers of people from Asia and Africa are seeking to enter the US illegally over the Mexican border.The vast majority of immigrants detained were from the Americas.However, a significant number were from Asian and African countries.We can obtain vertex dominating set by α-strong connections between these countries and vertex domination.In other words, We can find the countries which dominate others as α-strong from many countries which are increasing and they have a significant number.So We can study the main illegal immigration routes to the United States precisely, usefully and deeply.

Conclusion
At present, domination is considered to be one of the fundamental concepts in graph theory and its various applications to ad hoc networks, biological networks, distributed computing, social networks and web graphs partly explain the increased interest.
t-norm fuzzy graphs are the vast subject which have the fresh topics and many applications from the real-world problems that make the future better.So we defined domination which is a strong tools for analyzing data, on t-norm fuzzy graphs, for the first time.We hope this concept is useful for studying theoretical topics and applications on t-norm fuzzy graphs.

8 Preprints
and the degree of membership of a weakest edge is defined as its strength.If u 0 = u n and n ≥ 3 then P is called a cycle and P is called a 2/(www.preprints.org)| NOT PEER-REVIEWED | Posted: 10 April 2018 doi:10.20944/preprints201804.0119.v1 where N denotes the positive integers in Ref. ( [10], p. 131).

Example 3 . 3 .
Let (σ, µ) be a t-norm fuzzy graph with respect to ∧.By attention to it In Figure(1), the edges v 2 v 5 , v 2 v 4 , v 3 v 4 and v 1 v 3 are α-strong and the edges v 1 v 4 , v 1 v 2 and v 4 v 5 are not α-strong.Definition 3.4.Let ⊗ be a t-norm.Let (σ, µ) be a t-norm fuzzy graph with respect to ⊗.A subset S of V is called a α-strong dominating set in G if for every v ∈ S, there exists u ∈ S such that u dominates v as α-strong.Definition 3.5.Let G = (σ, µ) be a t-norm fuzzy graph with respect to a t-norm ⊗.Let S be the set of all α-strong dominating sets in G.The vertex domination number of G is defined as min D∈S [Σ u∈D (σ(u) + ds(u) d(u) )] and it is denoted by γ v (G).If d(u) = 0, for some u ∈ V, then we consider ds(u) d(u) equal with 0. The α-strong dominating set that is correspond to γ v (G) is called by vertex dominating set.We also say Σ u∈D (σ(u) + ds(u) d(u) ), vertex weight of D, for every D ∈ S and it is denoted by w v (D).Example 3.6.Let G = (σ, µ) be a t-norm fuzzy graph with respect to ∧.In Figure(1), the set {v 2 , v 3 } is the α-strong dominating set.This set is also vertex dominating set in t-norm fuzzy graph G with respect to ∧. Hence γ v (G) = 1.75 + 0.9 + 0.7 = 3.35.So γ v (G) = 3.35.Definition 3.7.Ref. [10], Definition 3.1, p. 131] Let (σ, µ) be a fuzzy graph with respect to ⊗.Then (σ, µ) is said to be complete with respect to ⊗, if for all

Figure 1 .
Figure 1.Vertex domination and ⊗ is defined as a t-norm ∧.Finally, Let V, σ, and µ be the vertices, value of vertices and value of edges respectively.In other words, By attention to fuzzy graph with respect to ∧ In Figure (1), the edges v 2 v 5 , v 2 v 4 , v 3 v 4 and v 1 v 3 are α-strong and the edges v 1 v 4 , v 1 v 2 and v 4 v 5 are not α-strong.So the set {v 2 , v 3 } is the α-strong dominating set.This set is also vertex dominating set in t-norm fuzzy graph G. Hence
such that u and v i are center and leaves of G, for 1 ≤ i ≤ n, respectively.The edge uv i , 1 ≤ i ≤ n is omly path between u and v i .So {u} is vertex dominating set in G. G is α-strong edgeless by Corollary(3.16).So