Some more results on fuzzy k-competition graphs

Fuzzy competition graph as the generalization of competition graph is introduced here. Fuzzy k-competition graph as a special type of fuzzy competition graph is defined here along with fuzzy isolated vertices. The fuzzy competition number is also introduced and investigated several properties. Isomorphism properties on fuzzy competition graphs are discussed.


I. INTRODUCTION
The concept of fuzzy graph was introduced by Rosenfeld [36] in 1975.Fuzzy graph theory has a vast area of applications.It is used in evaluation of human cardiac function, fuzzy neural networks, etc. Fuzzy graphs can be used to solve traffic light problem, time table scheduling, etc.In fuzzy set theory, there are different types of fuzzy graphs which may be a graph with crisp vertex set and fuzzy edge set or fuzzy vertex set and crisp edge set or fuzzy vertex set and fuzzy edge set or crisp vertices and edges with fuzzy connectivity, etc.A lot of works have been done on fuzzy graphs [5], [6], [14], [20], [22], [26].
In 1968, Cohen [12] introduced the notion of competition graphs in connection with a problem in ecology.Let − → D = (V, − → E ) be a digraph, which corresponds to a food web.A vertex x ∈ V ( − → D) represents a species in the food web and an arc ) 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( ) 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 . The competition graph is also applicable in channel assignment, coding, modelling of complex economic and energy systems, etc. [35].
A lot of works have been done on competition graphs and its variations.In all these works, it is assumed that the vertices and edges of the graphs are precisely defined.But, in reality we observe that sometimes the vertices and edges of a graph can not be defined precisely.For example, in ecology, species may be of different types like vegetarian, non-vegetarian, strong, weak, etc. Similarly in ecology, preys may be tasty, digestive, harmful, etc.The terms tasty, digestive, harmful, etc. have no precise meanings.They are fuzzy in nature and hence the species and preys may be assumed as fuzzy sets and inter-relationship between the species and preys can be designed by a fuzzy graph.This motivates the necessity of fuzzy competition graphs.
The competition graphs and fuzzy graphs are well known topics.In this article, fuzzy competition graphs are defined as motivated from fuzzy food web.Also, the generalization of it, the fuzzy k-competition graphs is introduced.Fuzzy neighbourhood graphs and their properties are investigated.Fuzzy competition number and fuzzy isolated vertex are exemplified.Isomorphism of fuzzy competition graphs is discussed.

A. Review of literature
In 1968, Cohen [12] presented a nice technique for food webs to find minimum number of dimensions of niche spaces and defined competition graphs.After Cohen's introduction of competition graph, several variations of it are found in literature.These are p-competition graphs of a digraph [16], [19], tolerance competition graphs [9], m-step competition graphs of a digraph [11], competition hypergraphs [46], common enemy graph of a digraph [21], competition-common enemy graph of a digraph [45], etc. Surveys of the large literature related to competition graphs can be found in [10], [15], [43].All these representations are crisp in sense and do not include all real field competitions.The competition graphs describe about the common prey and related species, but they do not measure the strength of competitions.These graphs do not show that how much the species depend on a common prey compared to other species.On the other hand fuzzy graph theory, after Rosenfeld [36], is increased with a large number of branches [2], [3], [4], [27], [28], [29], [30], [32], [34], [37], [38].
To include the representations of all real world competitions, fuzzy competition graphs are introduced.The relations among neighbourhoods of any species are described in fuzzy neighbourhood graphs.Sometimes the preys are so valuable that the species (competitors) compete strongly.Different level www.ijarai.thesai.org of competitions can shown by "fuzzy k-competition graphs", a generalization of fuzzy competition graph which is introduced here.

A. Competition graphs
) of ordered pairs of distinct vertices called arcs.We will often write The open neighbourhood of a vertex is the union of out-neighbourhood and in-neighbourhood of the vertex.A walk in  , E) which has the same vertex set V and has an edge between distinct two vertices x, y ∈ V if there exist a vertex a ∈ V and arcs Many variations of competition graph have been available in literature [9], [11], [16], [19], [46].One of the important graphs, known as p-competition graphs, is defined below .

B. Fuzzy graphs
A fuzzy set A on a set X is characterized by a mapping m : X → [0, 1], which is called the membership function.
The membership function of the intersection of two fuzzy sets A and B with membership functions m A and m B respectively is defined as the minimum of the two individual membership functions.
The family of all fuzzy subsets is denoted by F(x).The cardinality of a fuzzy set A = (X, m) is a positive real number c(A) or |A| is the sum of membership values of the elements of X [?].Now the fuzzy graph is defined below.
and µ is a symmetric fuzzy relation on σ.Here σ(x) and µ(x, y) represent the membership values of the vertex x and of the edge (x, y) in ξ.
Since µ is well defined, a fuzzy graph has no multiple edges.A loop at a vertex x in a fuzzy graph is represented by µ(x, x) = 0.The fuzzy set (V, σ) is called fuzzy vertex set of ξ and the elements of the fuzzy set are called fuzzy vertices.(V × V, µ) is called the fuzzy edge set of ξ and the elements of the fuzzy set are called fuzzy edge.An edge is non-trivial if µ(x, y) = 0.The fuzzy graph ξ = (V , τ, ν) is called a fuzzy subgraph [24] of ξ if τ (x) ≤ σ(x) for all x ∈ V and ν(x, y) ≤ µ(x, y) for all x, y ∈ V where V ⊂ V .
Like crisp digraphs there is fuzzy digraphs which are defined below.
Definition 4: [25] Directed fuzzy graph (fuzzy digraph) A homomorphism [27] between fuzzy graphs ξ and ξ is a map h : S → S which satisfies σ(x) ≤ σ (h(x)) for all x ∈ S and µ(x, y) ≤ µ (h(x), h(y)) for all x, y ∈ S where S is set of vertices of ξ and S is that of ξ .
A weak isomorphism [27] between fuzzy graphs is a bijective homomorphism h : S → S which satisfies σ(x) = σ (h(x)) for all x ∈ S.

III. FUZZY COMPETITION GRAPHS
Now, we come to our main objective of the paper, the fuzzy competition graph.Like crisp graph, fuzzy out-neighbourhood and fuzzy in-neighbourhood of a vertex in directed fuzzy graph are defined below.
Definition 5: Fuzzy out-neighbourhood of a vertex v of a directed fuzzy graph  Edge in fuzzy competition graphs indicates that there is a competition between two vertices (species) for at least one prey.So the strengths of the edges are important to characterize the competitions.Now an extension of fuzzy competition graph, called fuzzy k-competition graph is defined in the following.
Definition 7: Let k be a non-negative number.The fuzzy k- an undirected fuzzy graph ξ = (V, σ, µ) which has the same fuzzy vertex set as − → ξ and has a fuzzy edge between two vertices x, y So fuzzy k-competition graph is simply fuzzy competition graph when k = 0.An example of fuzzy 0.15-competition graph is given below.
where m is the membership value of the element a.Here, 5. If all the edges of a fuzzy digraph are strong, then all the edges of the corresponding fuzzy competition graph may not be strong.This result is illustrated below.Let us consider two vertices x, y with σ(x) = 0.3, σ(y) = 0.4 in a fuzzy digraph such that the vertices have a common prey z with σ(z) = 0. Let the corresponding fuzzy competition graph be C( Case 1: Let N + (x) ∩ N + (y) be a null set for all x, y ∈ V .Then there exist no edge in C( − → ξ ) between x and y.Case 2: N + (x) ∩ N + (y) is not a null set.Let N + (x) ∩ N + (y) = {(a 1 , m 1 ), (a 2 , m 2 ), . . ., (a z , m z )}, where m i , i = 1, 2, . . ., z are the membership values of a i , i = 1, 2, . . ., z, respectively.So We have seen that if height of intersection between two out neighbourhoods of two vertices of a fuzzy digraph is greater than 0.5, the edge between the two vertices in corresponding fuzzy competition graph is strong.This result is not true in corresponding fuzzy k-competition graph.A related result is presented below.

A. Fuzzy isolated vertex
Isolated vertex in crisp graph is a vertex which has no incident edge i.e. a vertex of degree 0. In fuzzy graph, degree of a vertex is the sum of the membership values of incident edges.If the degree of a vertex in fuzzy graph is zero approximately, then the vertex can be assumed as isolated vertex.The formal definition of fuzzy isolated vertex is given below.
Definition 8: Let ξ = (V, σ, µ) be a fuzzy graph and be a pre-assigned positive real number.A vertex x is said to be fuzzy isolated vertex if the degree of x is less than i.e. d(x) < .
An example of isolated vertex is given as follows.
Example 4: Let = 0.05.In Fig. 4, the degree of the vertex b is 0.01 + 0.03 = 0.04.As d(b) < , b is said to be fuzzy isolated vertex.By similar reason, a and c are fuzzy isolated vertices.
The fuzzy isolated vertices are not same like crisp isolated vertices.Fuzzy isolated vertices may have an edge of small strength.The definition of strong isolated vertices are given below.www.ijarai.thesai.orgDefinition 9: Let ξ = (V, σ, µ) be a fuzzy graph.A vertex x ∈ V is said to be strong isolated vertex in ξ if d(x) = 0.
In fuzzy graph, vertices which are not connected to any other vertices are called strong isolated vertices.So, strong isolated vertices are comparable with crisp isolated vertices.

B. Fuzzy competition number
Fuzzy competition number is associated with a fuzzy competition graph which is defined below.An important theorem related to strong isolated vertices is given below.
Theorem 4: A path of length m along with a strong isolated vertex may be a fuzzy competition graph of a fuzzy directed tree of m + 1 vertices.Proof.Let P m = (V, σ, µ) be a fuzzy path where V = {v 1 , v 2 , . . ., v m }.We take another vertex u which is distinct from the vertices of P m .So, the fuzzy path together with the strong isolated vertex has m + 1 vertices.Now, we construct the directed fuzzy graph whose fuzzy competition graph is P m ∪ {u}.Keeping the concept of fuzzy competition graph of a fuzzy digraph, we construct the directed fuzzy edges −−−→ (v i , u), i = 1, 2, . . ., m.Now this directed fuzzy graph is a directed fuzzy tree T m+1 of m+1 vertices.Also, C(T m+1 ) = P m ∪ {u}.Hence the result.
The above theorem can be extended as follows.Some disjoint paths along with a strong isolated vertex may be a fuzzy competition graph of a fuzzy directed tree.Let the disjoint paths be P m1 , P m2 , . . ., P mn , then the number of vertices of the corresponding directed fuzzy graph is From Theorem 4, it is easy to observe that the fuzzy competition number of a path is 1.Like crisp graph the fuzzy competition numbers can be found as follows.
The competition number of a fuzzy chordal graph which has no strong isolated vertex is 1.
If a fuzzy graph is triangle free, then the fuzzy competition number is equal to two more than the difference between the number of edges and number of vertices.The competition number of fuzzy complete graph is one.

C. Fuzzy neighbourhood graphs
The fuzzy open neighbourhood and fuzzy closed neighbourhood of a vertex in fuzzy graph are defined below.
Definition 11: Fuzzy open neighbourhood of a vertex v of a fuzzy graph ξ = (V, σ, µ) is the fuzzy set In this section, fuzzy open neighbourhood graphs are defined and then fuzzy closed neighbourhood graphs.Based on these fuzzy graphs fuzzy k-neighbourhood graphs of open and closed types are defined.
Definition 12: Let ξ = (V, σ, µ) be a fuzzy graph.Fuzzy open neighbourhood graph of ξ is a fuzzy graph N (ξ) = (V, σ, µ ) whose fuzzy vertex set is same as ξ and has a fuzzy edge between two vertices x and y ∈ V in N (ξ) if and only if N (x)∩N (y) is non-empty fuzzy set in ξ and µ : Definition 13: Let ξ = (V, σ, µ) be a fuzzy graph.Fuzzy closed neighbourhood graph of ξ is a fuzzy graph N [ξ] = (V, σ, µ ) whose fuzzy vertex set is same as ξ and has a fuzzy edge between two vertices x and y ∈ Definition 14: Let ξ = (V, σ, µ) be a fuzzy graph.Fuzzy (k)-neighbourhood graph (read as open fuzzy kneighbourhood graph) of ξ is a fuzzy graph N k (ξ) = (V, σ, µ ) whose vertex set is same as ξ and has an edge between two vertices x and y ∈ Definition 15: Let ξ = (V, σ, µ) be a fuzzy graph.Fuzzy [k]-neighbourhood graph (read as fuzzy closed kneighbourhood graph) of ξ is a fuzzy graph N k [ξ] = (V, σ, µ ) whose fuzzy vertex set is same as ξ and has a fuzzy edge between two vertices x and y ∈ . Also an underlying fuzzy graph has the same fuzzy vertex set as the directed fuzzy graph.Hence N k (U (ξ)) has the same fuzzy vertex set as − → ξ .Now we need to show that ν(x, y) = ν (x, y) for all x, y ∈ . Also an underlying fuzzy graph has the same fuzzy vertex set as the directed fuzzy graph.Hence N k [U (ξ)] has the same fuzzy vertex set as − → ξ .Now we need to show that ν(x, y) = ν (x, y) for all x, y ∈ V .As the directed fuzzy graph − → ξ has a loop at every vertex, out-neighbourhood of each vertex contains the vertex itself.Hence

IV. ISOMORPHISM IN FUZZY COMPETITION GRAPH
Isomorphism on fuzzy graphs are well known in literature.Here the isomorphism in fuzzy digraphs are introduced.A homomorphism of fuzzy digraphs ) for all x, y ∈ S where S is the set of vertices of − → ξ and S is that of − → ξ .A weak isomorphism between fuzzy digraphs is a bijective homomorphism h : S → S which satisfies σ(x) = σ (h(x)) for all x ∈ S.
A co-weak isomorphism between fuzzy digraphs is a bijective homomorphism h : S → S which satisfies µ An isomorphism between fuzzy graphs of fuzzy digraphs is a bijective homomorphism h : S → S which satisfies σ(x) = σ (h(x)) for all x ∈ S and µ Isomorphism between fuzzy graphs is an equivalence relation.But, if there is an isomorphism between two fuzzy www.ijarai.thesai.orggraph and one is fuzzy competition graph, then the other will be fuzzy competition graph.But, the corresponding digraphs may not be isomorphic.This result can be shown as follows.This remark can be illustrated from the following example.In Fig. 6(a) and Fig. 6(b), two fuzzy digraphs are shown.These fuzzy digraphs are not isomorphic as degree of vertices of these fuzzy digraphs are not same.But, the corresponding fuzzy competition graphs are isomorphic.
If two fuzzy digraphs are isomorphic, their corresponding fuzzy competition graphs must be isomorphic.This can established in the following theorem.V. CONCLUSIONS This study introduces fuzzy competition graphs and fuzzy kcompetition graph.In these fuzzy graphs, if there is a common out neighbourhood of two vertices, then there will be an edge between the vertices.Thus the fuzzy competition graph can be stated as 1-step fuzzy competition graph.In future, m-step fuzzy competition graphs can be investigated as an extension of this study.Besides, fuzzy isolated vertices and fuzzy competition numbers are introduced.The competition numbers of paths and several graphs are investigated.But, fuzzy competition number of any fuzzy graphs is not calculated here.This can be investigated in future.Isomorphism in fuzzy graphs is new in the research field.Isomorphism relation between two fuzzy competition graphs are established.Also, some fundamental results related to fuzzy competition graphs are presented.These results will be helpful in ecosystem, competitive markets, etc.

Fig. 1 .
Fig. 1.Example of fuzzy out-neighbourhood and in-neighbourhood of a vertex

7 :
for similar reason.Hence ν(x, y) = ν (x, y) for all x, y ∈ V .Similarly, a relation between fuzzy [k]-neighbourhood graph and fuzzy k-competition graph is established below.Theorem If the symmetric fuzzy digraph − → ξ has loop at every vertex, then C k ( − → ξ ) = N k [U (ξ)] where U(ξ) is the loop less fuzzy graph underlying − → ξ .Proof.Let a directed fuzzy graph be − → ξ = (V, σ, − → µ ) and the corresponding underlying loop less graph be U

Theorem 8 :ξ 1 and − → ξ 2 ξ 2 )ξ 1 )ξ 2 )→ ξ 1 )→ ξ 2 )
If two fuzzy digraphs − → are isomorphic, then C( − → ξ 1 ) and C( − → are isomorphic.Proof.Here two fuzzy digraphs − → ξ 1 and − → ξ 2 are isomorphic.Then there exists one to one correspondence between vertices and edges.Also the membership values of the vertices and edges are preserved.So the adjacency of edges are preserved.Then it is easy to observe that C( − → and C( − → are isomorphic.Similarly, we can prove that if two fuzzy digraphs − → ξ 1 and − → ξ 2 are isomorphic, then C k ( − and C k ( − are isomorphic. [1]il is a walk in which all arcs are distinct.A path is a walk in which all vertices are distinct.A pathx 1 , x 2 , ..., x k with k ≥ 3 is a cycle if x 1 = x k .For an undirected graph, open-neighbourhood [1] N (x) of the vertex x is the set of all vertices adjacent to x in the graph.Open neighbourhood graph[1]N (G) of G is a graph whose vertex set is same as G and has an edge between two vertices x and y in N (G) if and only if N (x)∩N (y) = φ in G.
a vertex set V with an edge between x and y in V if and only if, for some distinct vertices a 1 , a 2 , . . ., a p in