New concepts of vague competition graphs

Abstract

A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, vague competition graph and m-step competition graphs related to a vague graph are introduced. Likewise, some interesting theorems on them, which are related to the independent strong edges of the vague competition graphs are investigated.

Keywords

Vague graphs vague competition graphs vague out-neighbourhood

1 Introduction

The origin of graph theory started with the problem of Konigsberg bridge, in 1735. This problem lead to the concept of Eulerian graph. Euler studied the problem of Konigsberg bridge and constructed a structure that solves the problem called Eulerian graph. In 1840, Mobious gave the idea of complete graph and bipartite graph and Kuratowski proved that they are planar by means of recreational problems. At present, graph theoretical concepts are highly utilized by computer science applications. Especially in research areas of computer science including datamining, image segmentation, clustering, and networking. The introduction of fuzzy sets by Zadeh [29] in 1965 changed the face of science and technology to a great extent. Fuzzy set paved the way for a new philosophical thinking of ‘Fuzzy Logic’ which now, is an essential concept in artificial intelligence. The most important feature of a fuzzy set is that it consists of a class of objects that satisfy a certain (or several) property. For example, for a fuzzy set A, each object x has a membership degree of A, denoted as μ_A (x). Gau and Buehrer [6] proposed the concept of vague set by replacing the value of an element in a set with a subinterval of [1]. Namely, a true-membership function t_v (x) and a false membership function f_v (x) are used to describe the bounderies of the membership degree. The first definition of fuzzy graphs was proposed by Kafmann [13] in 1993, from Zadeh’s fuzzy relations [29–31]. But Rosenfeld [23] introduced another elaborated definition including fuzzy vertex, fuzzy edges, and several fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness and etc. Ramakrishna [15] introduced the concept of vague graphs and studied some of their properties. Akram et al. [2] defined the vague hypergraphs. Borzooei and Rashmanlou [8–10] investigated domination and new concepts in vague graphs. Rashmanlou et al.[16–22] introduced new concepts of bipolar fuzzy graphs, compelete interval-valued fuzzy graphs, antipodal interval-valued fuzzy graphs, balanced interval-valued fuzzy graphs and some properties of highly irregular interval-valued fuzzy graphs. Samanta and Pal [26–27] defined fuzzy tolerance graph and fuzzy k-competition and p-competition fuzzy graph. In this paper, we have introduced vague competition graph and m-step competition graphs related to a vague graph. Also, some interesting theorems on them, which are related to the independent strong edges of the vague competition graphs are investigated. For further details on the literature, readers may look in [1, 3–7, 24, 25, 28]. After introductory section, (Section 1), some basic definitions are given in Section 2. In Section 3, the concepts of vague competition graphs, m-step vague competition graphs are defined. Some properties are established. At last conclusion is given in Section 4.

2 Preliminaries

A directed graph (digraph) $\vec{G}$ is a graph which consists of non-empty finite set $V (\vec{G})$ of elements called vertices an a finite set $\vec{E} (\vec{G})$ of ordered pairs of distinct vertices called arcs. We will often write $\vec{G} = (V, \vec{E})$ . For an arc $(\vec{v_{i} v_{j}})$ , v_i is the tail and v_j is the head. The order (size) of $\vec{G}$ is the number of vertices (arcs) in $\vec{G}$ . The out-neighbourhood of a vertex v_i is the set $N^{+} (v_{i}) = {v_{j} \in V - v_{i} : (\vec{v_{i} v_{j}}) \in \vec{E}}$ . Similarly, the in-neighbourhood N^- (v_i) of a vertex v_i is the set ${v_{j} \in V - v_{i} : (\vec{v_{j} v_{i}}) \in \vec{E}}$ . The open neighbourhood of a vertex v_i is N⁺ (v_i) ∪ N^- (v_i). A walk in $\vec{G}$ is an alternating sequence $W = x_{1} \vec{e_{1}} x_{2} \vec{e_{2}} \dots x_{k - 1} \vec{e_{k}} x_{k}$ of vertices x_i and arcs $\vec{e_{i}}$ of $\vec{G}$ such that tail of $\vec{e_{i}}$ is x_i and head is x_i+1, for every i = 1, 2, ⋯ , k - 1. A walk is closed if x₁ = x_k. A path is a walk in which all vertices are distinct. A path x₁, x₂, ⋯ , x_k with k ≥ 3 is a cycle if x₁ = x_k.

In 1968, Cohen [11] introduced the notion of competition graphs in connection with a problem in ecology. Let $\vec{D} = (V, E)$ be a digraph, which corresponds to a food web. A vertex $x \in V (\vec{D})$ represents a species in the food web and an arc $(\vec{xs}) \in \vec{E} (\vec{D})$ means that x preys on the species s. If two species x and y have a common prey s, they will compete for the prey s. Based on this analogy, Cohen defined a graph which represents the relations of competition among the species in the food web. The competition graph $C (\vec{D})$ of a digraph $\vec{D} = (V, \vec{E})$ is an undirected graph G = (V, E) which has the same vertex set V and has an edge between two distinct vertices x, y ∈ V if there exists a vertex s ∈ V and arcs $(\vec{xs}), (\vec{ys}) \in \vec{E} (\vec{D})$ . We say that a graph G is a competition graph if there exists a digraph $\vec{G}$ such that $C (\vec{G}) = G$ .

Definition 2.1. [14] For a positive integer p, the p-competition graph $C_{p} (\vec{G})$ corresponding to the digraph $\vec{G}$ is defined to have a vertex set V with an edge between x and y in V if and only if, for some distinct vertices a₁, a₂, ⋯ , a_p in V, $(\vec{{xa}_{1}}), (\vec{{ya}_{1}}), (\vec{{xa}_{2}}), (\vec{{ya}_{2}}),$ ⋯, $(\vec{{xa}_{p}}), (\vec{{ya}_{p}})$ are arcs in $\vec{G}$ . If $\vec{G}$ is thought of as a food web whose vertices are the species in some ecosystem, then (x, y) is an edge of $C_{𝓅} (\vec{G})$ if and only if x and y have at least p common preys. So, $C_{1} (\vec{G})$ is the competition graph.

Definition 2.2. [12] A vague set A on the set X is characterized by a mapping t_A : X → [0, 1], which is called as a true membership function and f_A : X → [0, 1], which is called as a false membership function such that 0 ≤ t_A (x) + f_A (x) ≤1, for all x ∈ X. A vague set is denoted by A = (X, t_A, f_A).

Definition 2.3. [15] Let G^* = (V, E) be a crisp graph. A pair G = (A, B) is called a vague graph on a crisp graph G^* = (V, E), where A = (X, t_A, f_A) is a vague set on V and B = (X, t_B, f_B) is a vague set on E ⊆ V × V such that t_B (xy) ≤ min(t_A (x) , t_A (y)) and f_B (xy) ≥ max(f_A (x) , f_A (y)) for each edge xy ∈ E. A vague graph G = (A, B) is said to be complete if t_B (v_iv_j) = min {t_A (v_i) , t_A (v_j)} and f_B (v_iv_j) = max {f_A (v_i) , f_A (v_j)}, for all v_i, v_j ∈ V. If G^* = (V, E) is a directed graph, then the vague graph G = (A, B) on G^* is called a directed vague graph (or vague digraph) and denoted by $\vec{G} = (A, B)$ .

Definition 2.4. [12] The cardinality of a vague set A is denoted by |A| = (|A|_t, |A|_f), where |A|_t represents the sum of true membership values and |A|_f represents the sum of false membership values of all elements of A.

Definition 2.5. [12] The height of a vague set A = (X, t_A, f_A) is defined as $h (A) = (sup_{x \in X} t_{A} (x), inf_{x \in X} f_{A} (x)) = (h_{1} (A), h_{2} (A)) .$

3 Vague competition graphs

Competition graphs are an extremely useful tools in solving the combinatorial problems in different areas including geometry, algebra, topology, optimization, and computer science. They have become particularly important in the study of networks. Hence, in this section, we come to our main objective of the paper, the vague competition graph. Like a crisp graph, we define vague out-neighbourhood and vague in-neighbourhood of a vertex for a directed vague graphs below.

Definition 3.1. (i) Vague out-neighbourhood of a vertex v in a directed vague graph $\vec{G} = (A, B)$ is the vague set $N^{f} (v) = (X_{v}^{f}, t_{v}^{f}, f_{v}^{f})$ , where $X_{v}^{f} = {u | t_{B} (\vec{uv}) > 0 and f_{B} (\vec{uv}) > 0}$ and $t_{v}^{f} : X_{v}^{f} \to [0, 1]$ defined by $t_{v}^{f} (u) = t_{B} (\vec{vu})$ and $f_{v}^{f} : X_{v}^{f} \to [0, 1]$ defined by $f_{v}^{f} (u) = f_{B} (\vec{vu})$ .

(ii) Vague in-neighbourhood of a vertex v of a directed vague graph $\vec{G} = (A, B)$ is the vague set $N^{t} (v) = (X_{v}^{t}, t_{v}^{t}, f_{v}^{t})$ , where $X_{v}^{t} = {u | t_{B} (\vec{uv}) > 0 and f_{B} (\vec{uv}) > 0}$ and $t_{v}^{t} : X_{v}^{t} \to [0, 1]$ defined by $t_{v}^{t} (u) = t_{B} (\vec{uv})$ and $f_{v}^{t} : X_{v}^{t} \to [0, 1]$ defined by $f_{v}^{t} (u) = f_{B} (\vec{uv})$ .

Example 3.2. Let $\vec{G} = (A, B)$ be directed vague graph on V = {u₁, u₂, u₃, u₄}, which is defined as follows (see Table 1 and Fig. 1):

Then N^f (u₁) = {(u₂, 0.2, 0.5) , (u₃, 0.1, 0.7)} , N^f (u₃) = {(u₂, 0.3, 0.5) , (u₄, 0.2, 0.6)} N^f (u₄) = {(u₂, 0.1, 0.6)} N^t (u₃) = {(u₁, 0.1, 0.7)}

Now, we define a vague competition graph.

Definition 3.3. The vague competition graph $C (\vec{G})$ of a vague digraph $\vec{G} = (A, B)$ is an undirected vague graph G = (A, B) which has the same vague vertex set as in $\vec{G}$ and has a vague edge between two vertices x, y ∈ V in $C (\vec{G})$ if and only if N^f (x) ∩ N^f (y) is a non-empty vague set in $\vec{G}$ . The true membership and false membership values of the edge (x, y) in $C (\vec{G})$ are t_B (xy) = h₁ (N^f (x) ∩ N^f (y)) , f_B (xy) = h₂ (N^f (x) ∩ N^f (y))

The following example illustrates vague competition graph.

Example 3.4. Let $\vec{G} = (A, B)$ be a directed vague graph on V = {u₁, u₂, u₃, u₄}, which is defined as follows (see Table 2 and Fig. 2):

Since N^f (u₁) = {(u₂, 0.2, 0.5) , (u₃, 0.1, 0.7)} , N^f (u₂) = ∅ , N^f (u₃) = {(u₄, 0.2, 0.6)} , N^f (u₄) = {(u₂, 0, 1)} we get N^f (u₁) ∩ N^f (u₄) = {(u₂, 0, 1)}. Hence, there is an edge between u₁ and u₄ in vague competition graph $C (\vec{G})$ , where true membership and false membership values ar given by t_B (u₁u₄) =0 and f_B (u₁u₄) =1. For all other edges we have t_B = f_B = 0.

An edge of the vague digraph $\vec{G} = (A, B)$ is said to be independent strong if $\frac{1}{2} [t_{A} (x) \land t_{A} (y)] < t_{B} (\vec{xy},$ $\frac{1}{2} [f_{A} (x) \lor f_{A} (y)] > f_{B} (\vec{xy}) .$ This definition is used in the following theorem.

Theorem 3.5.Let $\vec{G} = (A, B)$ be a vague digraph. If N^f (x) ∩ N^f (y) contains only one element of $\vec{G}$ , then the edge (x, y) of $C (\vec{G})$ is independent strong if and only if $| [N^{f} (x) \cap N^{f} (y)] |_{t} > 0.5 | [N^{f} (x) \cap N^{f} (y)] |_{f} < 0.5$ Proof. Let N^f (x) ∩ N^f (y) = {(a, p, q)}, where p and q are the true membership value and false membership value of either the edge (xa) or the edge (ya). Here, | [N^f (x) ∩ N^f (y)] |_t = p = h₁ (N^f (x) ∩ N^f (y)) and | [N^f (x) ∩ N^f (y)] |_f = q = h₂ (N^f (x) ∩ N^f (y)). So, t_B (xy) = p and f_B (xy) = q. Therefore, the edge (xy) in $C (\vec{G})$ is independent strong if and only if p > 0.5 and q < 0.5. Hence, the edge (xy) of $C (\vec{G})$ is independent strong if and only if | [N^f (x) ∩ N^f (y)] |_t > 0.5 and | [N^f (x) ∩ N^f (y)] |_f < 0.5. □

The open neighbourhood and closed neighbourhood of a vertex in vague graph are defined below.

Definition 3.6. The open neighbourhood of a vertex v of a vague graph G = (A, B) is the vague set N (v) = (X_v, t_v, f_v) where X_v = {u| t_B (uv) >0 and f_B (uv) >0}, t_v : X_v → [0, 1] and f_v : X_v → [0, 1] defined by t_v (u) = t_B (uv) and f_v (u) = f_B (uv). For each vertex v ∈ V, the vague singleton set, $A_{v} = ({v}, t_{A}^{'}, f_{A}^{'})$ such that $t_{A}^{'} : {v} \to [0, 1]$ and $f_{A}^{'} : {v} \to [0, 1]$ defined by $t_{A}^{'} (v) = t_{A} (v)$ and $f_{A}^{'} (v) = f_{A} (v)$ . The vague closed neighbourhood of a vertex v is N [v] = N (v) ∪ A_v.

Definition 3.7. Let G = (A, B) be a vague graph.

The open-neighbourhood graph of G is a graph N (G) = (A, B′), whose vertex set is same as G and has an edge between two vertices x and y ∈ V in N (G) if and only if N (x) ∩ N (y) is non-empty vague set in G. The true membership and false membership values of the edge xy are given by $t_{B}^{'} (xy) = h_{1} (N (x) \cap N (y))$ and $f_{B}^{'} (xy) = h_{2} (N (x) \cap N (y))$ .

The closed-neighbourhood graph of G is a graph N [G] = (A, B′), whose vertex set is same as G and has an edge between two vertices x and y ∈ V in N [G] if and only if N [x] ∩ N [y] is non-empty set in G. The true membership and false membership values of the edge xy are given by $t_{B}^{'} (xy) = h_{1} (N [x] \cap N [y])$ and $f_{B}^{'} (xy) = h_{2} (N [x] \cap N [y])$ .

Example 3.8. Let G = (A, B) be a vague graph and V = {x, y, z, w} (see Table 3).

The corresponding vague open and closed neighbourhood graphs are shown in Fig. 3.

Theorem 3.9. For every edge of a vague graph G, there exists an edge in N [G].

Proof. Let ab be an edge of a vague graph G = (A, B). Let N [G] = (A, B′) be the corresponding closed neighbourhood of a vague graph. Let a, b ∈ N [a] and a, b ∈ N [b]. Then a, b ∈ N [a] ∩ N [b]. Hence, h₁ (N [a] ∩ N [b]) ≠0 and h₂ (N [a] ∩ N [b]) ≠0. So, $t_{B}^{'} (ab) = h_{1} (N [a] \cap N [b]) \neq 0$ and $t_{B}^{'} (ab) = h_{2} (N [a] \cap N [b]) \neq 0$ . Hence, for every edge ab in vague graph G, there exists an edge ab in N [G]. □

Here, another extension of a vague competition graph is considered called the m-step competition graph. Note that the following notations are used:

${\vec{p}}_{v, u}^{m} :$ A directed path of length m from v to u.

$N_{m}^{f} (v)$ : m-step out-neighbourhood of the vertex v.

$N_{m}^{t} (v)$ : m-step in-neighbourhood of the vertex v.

$C_{m} (\vec{G})$ : m-step competition graph of the digraph $\vec{G}$ .

Definition 3.10. Let $\vec{G} = (A, B)$ be a vague digraph. The m-step digraph of $\vec{G}$ is denoted by ${\vec{G}}_{m} = (A, B)$ where vertex set of $\vec{G}$ is same with the vague vertex set of ${\vec{G}}_{m}$ and edge (x, y) exist in ${\vec{G}}_{m}$ if there exist a directed path ${\vec{p}}_{x, y}^{m}$ in $\vec{G}$ .

Definition 3.11. (i) The m-step out-neighbourhood of a vertex v of a vague digraph $\vec{G}$ is the set $N_{m}^{f} (v) = (X_{v}^{f}, ρ^{f} (v), ζ^{f} (v))$ where $X_{v}^{f} = {u |$ there exists a directed path of length m from v to u, ${\vec{p}}_{v, u}^{m}}$ , $ρ^{f} (v) : X_{v}^{f} \to [0, 1]$ , $ζ^{f} (v) : X_{v}^{f} \to [0, 1]$ and defined by ρ^f (v) = $min {t_{B} (\vec{xy}),$ (xy) is an edge of ${\vec{p}}_{v, u}^{m}}$ , $ζ^{f} (v) = max {f_{B} (\vec{xy}),$ (xy) is an edge of ${\vec{p}}_{v, u}^{m}}$ .

(ii) The m-step in-neighbourhood of a vertex v of a vague digraph $\vec{G}$ is the set $N_{m}^{t} (v) = (X_{v}^{f}, ρ^{t} (v), ζ^{t} (v))$ where $X_{v}^{t} = {u |$ there exists a directed path of length m from u to v, ${\vec{p}}_{u, v}^{m}}$ , $ρ^{t} (v) : X_{v}^{t} \to [0, 1]$ and $ζ^{t} (v) : X_{v}^{t} \to [0, 1]$ and defined by ρ^t (v) = $min {t_{B} (\vec{xy}),$ (xy) is an edge of ${\vec{p}}_{u, v}^{m}}$ , $ζ^{t} (v) = max {f_{B} (\vec{xy}),$ (xy) is an edge of ${\vec{p}}_{u, v}^{m}}$ .

Definition 3.12. Let $\vec{G} = (A, B)$ be a directed vague graph. The m-step competition graph of the vague digraph $\vec{G}$ is denoted by $C_{m} (\vec{G}) = (A, B)$ which has the same vertex set as in $\vec{G}$ and has an edge between two vertices x, y ∈ V in C_m (G) if and only if $(N_{m}^{f} (x) \cap N_{m}^{f} (y))$ is a non-empty set in $\vec{G}$ . The true membership and false membership values of the edge (xy) in $C (\vec{G})$ are given by $t_{B} (xy) = h_{1} (N_{m}^{f} (x) \cap N_{m}^{f} (y))$ and $f_{B} (xy) = h_{2} (N_{m}^{f} (x) \cap N_{m}^{f} (y))$ .

Example 3.13. Let $\vec{G} = (A, B)$ be a vague digraph on V = {a, b, c, d, e, f}, which is defined as followes (see Table 4):

Then, $N_{2}^{f} (x) = {(b, 0.1, 0.5), (c, 0.1, 0.6)}$ and $N_{2}^{f} (y)$ = {(b, 0.1, 0.7) , (c, 0.1, 0.7)}. So, $N_{2}^{f} (x) N_{2}^{f} (y) = {(b, 0.1, 0.7), (c, 0.1, 0.7)}$ . Hence t_B (xy) =0.1 and f_B (xy) =0.7. It is shown in Fig. 4.

Definition 3.14.m-step neighbourhood of a vertex v of a vague graph G = (A, B) is the set N_m (v) = (X_v, ρ (v) , ζ (v)) where X_v = {u| there exists a directed path of length m from v tou, $p_{v, u}^{m}}$ , ρ (v) : X_v → [0, 1] and ζ (v) : X_v → [0, 1] and defined by ρ (v) = min {t_B (xy) , (xy) is an edge of $p_{v, u}^{m}}$ , $ζ (v) = max {f_{B} (xy), (xy) is an edge of p_{v, u}^{m}}$ .

Definition 3.15. Let G = (A, B) be a vague graph. The m-step neighbourhood graph is denoted by N_m (G) and defined by N_m (G) = (A, C) where A = (t_A, f_A), C = (t_C, f_C), t_C : V × V → [0, 1] and f_C : V × V → [0, 1] are such that t_C (xy) = h₁ (N_m (x) ∩ N_m (y)) and f_C (xy) = h₂ (N_m (x) ∩ N_m (y)).

Note that the underlying vague graph of $\vec{G}$ is the vague graph G = (A, B), where A = (t_A, f_A), B = (t_B, f_B) where $t_{B} (xy) = min {t_{B} (\vec{xy}), t_{B} (\vec{yx})}$ and $f_{B} (xy) = max {f_{B} (\vec{xy}), f_{B} (\vec{yx})}$ , for all x, y ∈ V.

Theorem 3.16. Let $\vec{G} = (A, B)$ be a directed vague graph. If m > |V| then, $C_{m} (\vec{G})$ has no edges.

Proof. Let $\vec{G} = (A, B)$ be a vague digraph and $C_{m} (\vec{G}) = (A, B)$ be the corresponding m-step competition graph, where $t_{B} (xy) = h_{1} (N_{m}^{f} (x) \cap N_{m}^{f} (y))$ and $f_{B} (xy) = h_{2} (N_{m}^{f} (x) \cap N_{m}^{f} (y))$ . If m > |V|, there does not exist any directed path of length m in $\vec{G}$ . So, $(N_{m}^{f} (x) \cap N_{m}^{f} (y)$ is a non-empty set. Then, t_B (xy) =0 and f_B (xy) =0, for all $(xy) \in C_{m} (\vec{G})$ . Hence, $C_{m} (\vec{G})$ has no edges. □

Theorem 3.17. If $\vec{G}$ is a vague digraph and ${\vec{G}}_{m}$ is the m-step digraph of $\vec{G}$ , then $C ({\vec{G}}_{m}) = C_{m} (\vec{G})$ .

Proof. Clearly the vague vertex set of $C_{m} (\vec{G})$ and $C ({\vec{G}}_{m})$ are equal. Now we shall prove that the edge set of $C_{m} (\vec{G})$ and $C ({\vec{G}}_{m})$ are equal. Let (xy) be an edge in $C ({\vec{G}}_{m})$ . So, there exists directed edges $(\vec{{xa}_{1}}), (\vec{{ya}_{1}}), (\vec{{xa}_{2}}), (\vec{{ya}_{2}}), \dots, (\vec{{xa}_{k}}), (\vec{{ya}_{k}})$ for some positive integer k in ${\vec{G}}_{m}$ . Now, in ${\vec{G}}_{m}$ , N^f (x) ∩ N^f (y) = {(a_i, m_i, n_i) | i = 1, 2, ⋯ , k} where $m_{i} = t_{C} (\vec{{xa}_{i}}) \land t_{C} (\vec{{ya}_{i}})$ and $n_{i} = f_{C} (\vec{{xa}_{i}}) \lor f_{C} (\vec{{ya}_{i}})$ . Let M = max {m_i| i = 1, 2, ⋯ , k} and N = min {n_i| i = 1, 2, ⋯ , k}. Then, $\begin{matrix} t_{C} (xy) = h_{1} (N^{f} (x) \cap N^{f} (y)) = M \\ f_{C} (xy) = h_{2} (N^{f} (x) \cap N^{f} (y)) = N \end{matrix}$ An edge $(\vec{{xa}_{i}})$ exists in ${\vec{G}}_{m}$ that implies there exists a vague directed path from x to a_i of length m, ${\vec{p}}_{(x, a_{i})}^{m}$ in $\vec{G}$ and $\begin{matrix} t_{C} (\vec{{xa}_{i}}) = min {t_{B} (\vec{uv}) | (\vec{uv}) is an edge in {\vec{p}}_{x, a_{i}}^{m}} \\ f_{C} (\vec{{xa}_{i}}) = max {f_{B} (\vec{uv}) | (\vec{uv}) is an edge in {\vec{p}}_{x, a_{i}}^{m}} \end{matrix}$

Thus, the edge (x, y) is also the edge of $C_{m} (\vec{G})$ . Let $h_{1} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = M$ and $h_{2} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = N$ in $\vec{G}$ . Therefore, $t_{B} (xy) = h_{1} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = M$ and $f_{B} (xy) = h_{2} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = N$ . This proves that there exists an edge in $C_{m} (\vec{G})$ for every edge in $C ({\vec{G}}_{m})$ . Similarly, for every edge in $C_{m} (\vec{G})$ there exists an edge in $C ({\vec{G}}_{m})$ . This proves that $C ({\vec{G}}_{m}) = C_{m} (\vec{G})$ . □

Theorem 3.18. If vague digraph $\vec{G}$ does not contains any parallel edge, then $C_{m} (\vec{G}) = N_{m} (G)$ , where for m > 1 and G is the underlying vague graphof $\vec{G}$ .

Proof. Let $\vec{G} = (A, B)$ be a vague digraph contains no parallel edges. Then t_B (xy) = $min {t_{B} (\vec{xy}),$ $t_{B} (\vec{yx})}$ and $f_{B} (xy) = max {f_{B} (\vec{xy}), f_{B} (\vec{yx})}$ , for all x, y ∈ V. Since the vague digraph contains no parallel edges. Then, $t_{B} (xy) = t_{B} (\vec{xy})$ and $f_{B} (xy) = f_{B} (\vec{xy})$ , for all x, y ∈ V. Also, let $C_{m} (\vec{G}) = (A, C)$ be the m-step competition graph of $\vec{G}$ . Also, let N_m (G) = (A, B) be the m-step neighbourhood graph of G. Clearly the vertex set of $C_{m} (\vec{G})$ and N_m (G) are same. Now, we shall prove that the set of edges of $C_{m} (\vec{G})$ and N_m (G) are same. Since $t_{B} (xy) = t_{B} (\vec{xy})$ and $f_{B} (xy) = f_{B} (\vec{xy})$ , for all x, y ∈ V. Then, $h_{1} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = h_{1} (N_{m} (x) \cap N_{m} (y))$ and $h_{2} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = h_{2} (N_{m} (x) \cap N_{m} (y))$ . Therefore, $t_{C} (xy) = h_{1} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = h_{1} (N_{m} (x) \cap N_{m} (y)) = t_{D} (xy)$ and $f_{C} (xy) = h_{2} (N_{m}^{f} (x) \cap N_{m}^{f} (y)) = h_{2} (N_{m} (x) \cap N_{m} (y)) = f_{D} (xy)$ . This prove that the set of edges of $C_{m} (\vec{G})$ and N_m (G) are same. Hence, $C_{m} (\vec{G}) = N_{m} (G)$ . □

4 Conclusion

It is well known that graphs are among the most ubiquitous models of both natural and human-made structure. They can be used to model many types of relations and process dynamics in computer science, physical, biological, and social systems. Many problems of practical interest can be represented by graphs. In this paper, we have described vague competition graphs and m-step competition graphs of a vague graph. Also, we have established some theorems on them, which are related to the independent strong edges of the vague competition graphs. In future, Vague k-competition graphs will be introduced based on the analogy of this paper.

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