Intervalvalued fuzzy \(\phi\)tolerance competition graphs
 Tarasankar Pramanik†^{1},
 Sovan Samanta†^{2},
 Madhumangal Pal^{3},
 Sukumar Mondal^{4} and
 Biswajit Sarkar^{2}Email author
Received: 5 April 2016
Accepted: 5 October 2016
Published: 15 November 2016
Abstract
This paper develops an intervalvalued fuzzy \(\phi\)tolerance competition graphs which is the extension of basic fuzzy graphs and \(\phi\) is any real valued function. Intervalvalued fuzzy \(\phi\)tolerance competition graph is constructed by taking all the fuzzy sets of a fuzzy \(\phi\)tolerance competition graph as intervalvalued fuzzy sets. Product of two IVFPTCGs and relations between them are defined. Here, some hereditary properties of products of intervalvalued fuzzy \(\phi\)tolerance competition graphs are represented. Application of intervalvalued fuzzy competition graph in image matching is given to illustrate the model.
Keywords
Competition Tolerance Intervalvalued fuzzy graphsBackground
Graphs can be considered as the bonding of objects. To emphasis on a real problem, those objects are being bonded by some relations such as, friendship is the bonding of pupil. If the vagueness in bonding arises, then the corresponding graph can be modelled as fuzzy graph model. There are many research available in literature like Bhutani and Battou (2003) and Bhutani and Rosenfeld (2003).
Competition graph was defined in Cohen (1968). In ecology, there is a problem of food web which is modelled by a digraph \(\overrightarrow{D}=(V,\overrightarrow{E})\). In food web there is a competition between species (members of food web). A vertex \(x\in V(\overrightarrow{D})\) represents a species in the food web and arc \(\overrightarrow{(x,s)}\in \overrightarrow{E}(\overrightarrow{D})\) means that x kills the species s. If two species x and y have common prey s, they will compete for s. Based on this analogy, Cohen (1968) defined a graph model (competition graph of a digraph), which represents the relationship of competition through the species in the food web. The corresponding undirected graph \(G=(V,E)\) of a certain digraph \(\overrightarrow{D}=(V, \overrightarrow{E})\) is said to be a competition graph \(C(\overrightarrow{D})\) with the vertex set V and the edge set E, where \((x,y)\in E\) if and only if there exists a vertex \(s\in V\) such that \(\overrightarrow{(x,s)},\overrightarrow{(y,s)}\in \overrightarrow{E(\overrightarrow{D})}\) for any \(x,y\in V,\, (x\ne y)\).
There are several variations of competition graphs in Cohen’s contribution (Cohen 1968). After Cohen, some derivations of competition graphs have been found in Cho et al. (2000). In that paper, mstep competition graph of a digraph was defined. The pcompetition graph of a digraph is defined in Kim et al. (1995). The pcompetition means if two species have at least pcommon preys, then they compete to each other.
In graph theory, an intersection graph is a graph which represents the intersection of sets. An interval graph is the intersection of multiset of intervals on real line. Interval graphs are useful in resource allocation problem in operations research. Besides, interval graphs are used extensively in mathematical modeling, archaeology, developmental psychology, ecological modeling, mathematical sociology and organization theory.
Tolerance graphs were originated in Golumbic and Monma (1982) to extend some of the applications associated with interval graphs. Their original purpose was to solve scheduling problems for arrangements of rooms, vehicles, etc. Tolerance graphs are generalization of interval graphs in which each vertex can be represented by an interval and a tolerance such that an edge occurs if and only if the overlap of corresponding intervals is at least as large as the tolerance associated with one of the vertices. Hence a graph \(G = (V,E)\) is a tolerance graph if there is a set \(I = \{I_v{:}\,v \in V\}\) of closed real intervals and a set \(\{T_v{:}\,v \in V\}\) of positive real numbers such that \((x,y) \in E\) if \(I_x\cap I_y \ge {{\rm min}} \{ T_x,T_y\}\). The collection <\(I,T\)> of intervals and tolerances is called tolerance representation of the graph G.
Tolerance graphs were used in order to generalize some well known applications of interval graphs. In Brigham et al. (1995), tolerance competition graphs was introduced. Some uncertainty is included in that paper by assuming tolerances of competitions. A recent work on fuzzy kcompetition graphs is available in Samanta and Pal (2013). In the paper, fuzziness is applied in the representation of competitions. Recently Pramanik et al. defined and studied fuzzy \(\phi\)tolerance competition graph in Pramanik et al. (2016). But, fuzzy phitolerance targets only numbers between 0 and 1, but intervalvalued numbers are more appropriate for uncertainty. Other many related works are found in Pramanik et al. (2014) and Samanta and Pal (2015).
Contributions of the authors towards interval valued \(\phi\)tolerance competition graphs
Authors  Year  Contributions 

Cohen (1968)  1968  Introduced competition graphs 
Kauffman (1973)  1973  Defined fuzzy graphs 
Rosenfeld (1975)  1975  Modified the concept of fuzzy graphs given by Kauffman (1973) 
Golumbic and Monma (1982)  1982  Established the concept of tolerance graphs 
Cho et al. (2000)  2000  Defined mstep competition graphs 
Samanta and Pal (2011)  2011  Introduced fuzzy tolerance graphs 
Samanta and Pal (2013)  2013  Proposed the concept of fuzzy competition graphs 
Pramanik et al. (2016)  2016  Advanced the idea of fuzzy \(\phi\)tolerance competition graphs and defined \(\phi\)tolerance competition graphs 
This paper  –  Introduction of interval valued fuzzy \(\phi\)tolerance competition graphs 
Preliminaries
A function \(\alpha {:}\,X\rightarrow [0,1]\), called the membership function defined on the crisp set X is said to be a fuzzy set \(\alpha\) on X. The support of \(\alpha\) is \({{\mathrm{supp}}}(\alpha ) =\{x\in X \alpha (x)\ne 0\}\) and the core of \(\alpha\) is \({\mathrm{core}}(\alpha ) = \{x\in X \alpha (x)=1\}\). The support length is \(s(\alpha )={{\mathrm{supp}}}(\alpha )\) and the core length is \(c(\alpha )={{\mathrm{core}}}(\alpha )\). The height of \(\alpha\) is \(h(\alpha ) =\max \{\alpha (x) x\in X\}\). The fuzzy set \(\alpha\) is said to be normal if \(h(\alpha )=1\).
A fuzzy graph with a nonvoid finite set V is a pair \(G = (V, \sigma ,\mu )\), where \(\sigma {:}\,V \rightarrow [0,1]\) is a fuzzy subset of V and \(\mu {:}\,V\times V\rightarrow [0,1]\) is a fuzzy relation (symmetric) on the fuzzy subset \(\sigma\), such that \(\mu (x,y) \le \sigma (x) \wedge \sigma (y)\), for all \(x,y\in V\), where \(\wedge\) stands for minimum. The degree of a vertex v of a fuzzy graph \(G = (V, \sigma ,\mu )\) is \(\displaystyle d(v)=\sum \nolimits _{u\in V\{v\}}\mu (v,u)\). The order of a fuzzy graph G is \(\displaystyle O(G)=\sum \nolimits _{u\in V}\sigma (u)\). The size of a fuzzy graph G is \(\displaystyle S(G)=\sum \mu (u,v)\).
Let \({\mathcal {F}}=\{\alpha _1,\alpha _2,\ldots , \alpha _n\}\) be a finite family of fuzzy subsets on a set X. The fuzzy intersection of two fuzzy subsets \(\alpha _1\) and \(\alpha _2\) is a fuzzy set and defined by \(\alpha _1\wedge \alpha _2=\left\{ \min \{\alpha _1(x),\alpha _2(x)\}x\in X\right\}\). The union of two fuzzy subsets \(\alpha _1\) and \(\alpha _2\) is a fuzzy set and is defined by \(\alpha _1\vee \alpha _2=\left\{ \max \{\alpha _1(x),\alpha _2(x)\}x\in X\right\}\). \(\alpha _1\le \alpha _2\) for two fuzzy subsets \(\alpha _1\) and \(\alpha _2\), if \(\alpha _1(x)\le \alpha _2(x)\) for each \(x\in X\).
Let us consider a family of fuzzy intervals \({\mathcal {F}}_{\mathcal {I}}=\{{\mathcal {I}}_1, {\mathcal {I}}_2, \ldots , {\mathcal {I}}_n\}\) on X. Then the fuzzy interval graph is the fuzzy intersection graph of these fuzzy intervals \({\mathcal {I}}_1, {\mathcal {I}}_2, \ldots , {\mathcal {I}}_n\).
Fuzzy tolerance of a fuzzy interval is denoted by \({\mathcal {T}}\) and is defined by an arbitrary fuzzy interval, whose core length is a positive real number. If the real number is taken as L and \(i_ki_{k1}=L\), where \(i_k,i_{k1}\in R\), a set of real numbers, then the fuzzy tolerance is a fuzzy set of the interval \([i_{k1},i_k]\).
Fuzzy interval digraph is a directed fuzzy interval graph, whose edge membership function need not to be symmetric.
 (1)
\(D_1+D_2=[a_1^,a_1^+]+[a_2^,a_2^+]=[a_1^+a_2^ a_1^\cdot a_2^, a_1^+ +a_2^+  a_1^+\cdot a_2^+],\)
 (2)
\(\min \{D_1,D_2\}=[\min \{a_1^,a_2^\}, \min \{a_1^+,a_2^+\}],\)
 (3)
\(\max \{D_1,D_2\}=[\max \{a_1^,a_2^\}, \max \{a_1^+,a_2^+\}],\)
 (4)
\(D_1\le D_2 \Leftrightarrow a_1^\le a_2^\) and \(a_1^+\le a_2^+\),
 (5)
\(D_1=D_2 \Leftrightarrow a_1^= a_2^\) and \(a_1^+= a_2^+\),
 (6)
\(D_1<D_2 \Leftrightarrow D_1\le D_2\) and \(D_1\ne D_2\),
 (7)
\(kD_1=[ka_1^, ka_2^+]\), where \(0\le k\le 1\).
Fuzzy outneighbourhood of a vertex \(v\in V\) of an intervalvalued fuzzy directed graph (IVFDG) \(\overrightarrow{D}=(V,A,\overrightarrow{B})\) is the IVFS \({\mathcal {N}}^+(v)=(X_v^+, m_v^+)\), where \(X_v^+=\{u{:}\, \mu _B(\overrightarrow{v,u})>0\}\) and \(m_v^+{:}\,X_v^+\rightarrow [0,1]\times [0,1]\) defined by \(m_v^+=\mu _B(\overrightarrow{v,u})=[\mu _B^(\overrightarrow{v,u}), \mu _B^+(\overrightarrow{v,u})]\)
An intervalvalued fuzzy graph of a graph \(G^*=(V,E)\) is a fuzzy graph \(G=(V, A, B)\), where \(A=[\mu _A^, \mu _A^+]\) is an intervalvalued fuzzy set on V and \(B=[\mu _B^, \mu _B^+]\) is a symmetric intervalvalued fuzzy relation on E. An intervalvalued fuzzy digraph \(\overrightarrow{G}=(V, A, \overrightarrow{B})\) is an intervalvalued fuzzy graph, where the fuzzy relation \(\overrightarrow{B}\) is antisymmetric.
An intervalvalued fuzzy graph \(\xi = (A,B)\) is said to be complete intervalvalued fuzzy graph if \(\mu ^(x,y)= \min \{\sigma ^(x),\sigma ^(y)\}\) and \(\mu ^+(x,y)=\) \(\min\) \(\{\sigma ^+(x),\) \(\sigma ^+(y)\}\), \(\forall x,y\in V\). An intervalvalued fuzzy graph is defined to be bipartite, if there exists two sets \(V_1\) and \(V_2\) such that the sets \(V_1\) and \(V_2\) are partitions of the vertex set V, where \(\mu ^+(u,v)=0\) if \(u,v\in V_1\) or \(u, v \in V_2\) and \(\mu ^+(v_1, v_2) > 0\) if \(v_1\in V_1\) (or \(V_2\)) and \(v_2 \in V_2\) (or \(V_1\)).
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1\times A_2}^(x_1, x_2) = \min \{\mu _{A_1}^(x_1), \mu _{A_2}^(x_2)\}\\ \mu ^+_{A_1\times A_2}(x_1, x_2) = \min \{\mu ^+_{A_1}(x_1), \mu ^+_{A_2}(x_2)\} \end{array}\right\}\) for all \(x_1\in V_1, x_2\in V_2\),
 (2)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^((x,x_2),(x,y_2)) = \min \{\mu _{A_1}^(x), \mu _{B_2}^(x_2,y_2)\}\\ \mu _{B_1\times B_2}^+((x,x_2),(x,y_2)) = \min \{\mu _{A_1}^+(x), \mu _{B_2}^+(x_2,y_2)\} \end{array}\right\}\) for all \(x\in V_1\) and \((x_2, y_2)\in E_2\),
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^((x_1,y),(y_1,y)) = \min \{\mu _{B_1}^(x_1,y_1), \mu _{A_2}^(y)\}\\ \mu _{B_1\times B_2}^+((x_1,y),(y_1,y)) = \min \{\mu _{B_1}^+(x_1,y_1), \mu _{A_2}^+(y)\} \end{array}\right\}\) for all \((x_1,y_1)\in E_1\) and \(y \in V_2.\)
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1\circ A_2}^(x_1, x_2) = \min \{\mu _{A_1}^(x_1), \mu _{A_2}^(x_2)\}\\ \mu ^+_{A_1\circ A_2}(x_1, x_2) = \min \{\mu ^+_{A_1}(x_1), \mu ^+_{A_2}(x_2)\} \end{array}\right\}\) for all \(x_1\in V_1, x_2\in V_2\),
 (2)
\(\left\{ \begin{array}{l} \mu _{B_1\circ B_2}^((x,x_2),(x,y_2)) = \min \{\mu _{A_1}^(x), \mu _{B_2}^(x_2,y_2)\}\\ \mu _{B_1\circ B_2}^+((x,x_2),(x,y_2)) = \min \{\mu _{A_1}^+(x), \mu _{B_2}^+(x_2,y_2)\} \end{array}\right\}\) for all \(x\in V_1\) and \((x_2, y_2)\in E_2\),
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1\circ B_2}^((x_1,y),(y_1,y)) = \min \{\mu _{B_1}^(x_1,y_1), \mu _{A_2}^(y)\}\\ \mu _{B_1\circ B_2}^+((x_1,y),(y_1,y)) = \min \{\mu _{B_1}^+(x_1,y_1), \mu _{A_2}^+(y)\} \end{array}\right\}\) for all \((x_1,y_1)\in E_1\) and \(y \in V_2,\)
 (4)
\(\left\{ \begin{array}{l} \mu _{B_1\circ B_2}^((x_1,x_2),(y_1,y_2)) = \min \{\mu _{A_2}^(x_2), \mu _{A_2}^(y_2),\mu _{B_1}^(x_1,y_1)\}\\ \mu _{B_1\circ B_2}^+((x_1,x_2),(y_1,y_2)) = \min \{\mu _{A_2}^+(x_2), \mu _{A_2}^+(y_2),\mu _{B_1}(x_1,y_1)\} \end{array}\right\}\) otherwise.
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1\cup A_2}^(x) =\mu _{A_1}^(x) {\text { if }}\,x\in V_1 {\text { and }}\, x\notin V_2\\ \mu _{A_1\cup A_2}^(x) =\mu _{A_2}^(x) {\text { if }}\,x\in V_2 {\text { and }}\,x\notin V_1\\ \mu _{A_1\cup A_2}^(x) =\max \{\mu _{A_1}^(x), \mu _{A_2}^(x)\}\,{\text { if }}\,x\in V_1\cap V_2. \end{array}\right.\)
 (2)
\(\left\{ \begin{array}{l} \mu _{A_1\cup A_2}^+(x) =\mu _{A_1}^+(x) {\text { if }}\, x\in V_1 {\text { and }}\,x\notin V_2\\ \mu _{A_1\cup A_2}^+(x) =\mu _{A_2}^+(x) {\text { if }}\,x\in V_2 {\text { and }}\,x\notin V_1\\ \mu _{A_1\cup A_2}^+(x) =\max \{\mu _{A_1}^+(x), \mu _{A_2}^+(x)\} {\text { if }}\,x\in V_1\cap V_2. \end{array}\right.\)
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^(x,y) = \mu _{B_1}^(x,y) {\text { if }}\,(x,y)\in E_1 {\text{and}}\,(x,y)\notin E_2\\ \mu _{B_1\times B_2}^(x,y) = \mu _{B_2}^(x,y) {\text{if}}\,(x,y)\in E_2 {\text{and}}\,(x,y)\notin E_1\\ \mu _{B_1\times B_2}^(x,y) = \max \{\mu _{B_1}^(x,y), \mu _{B_2}^(x,y)\} {\text{if}}\,(x,y)\in E_1\cap E_2. \end{array}\right.\)
 (4)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^+(x,y) = \mu _{B_1}^+(x,y) {\text{if}}\,(x,y)\in E_1 {\text{and}}\,(x,y)\notin E_2\\ \mu _{B_1\times B_2}^+(x,y) = \mu _{B_2}^+(x,y) {\text{if}}\,(x,y)\in E_2 {\text{and}}\,(x,y)\notin E_1\\ \mu _{B_1\times B_2}^+(x,y) = \max \{\mu _{B_1}^+(x,y), \mu _{B_2}^+(x,y)\} {\text{if}}\,(x,y)\in E_1\cap E_2. \end{array}\right.\)
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1+ A_2}^(x) = (\mu _{A_1}^\cup \mu _{A_2}^)(x)\\ \mu _{A_1+ A_2}^+(x) = (\mu _{A_1}^+\cup \mu _{A_2}^+)(x) \end{array}\right\}\) if \(x\in V_1\cup V_2\),
 (2)
\(\left\{ \begin{array}{l} \mu _{B_1+ B_2}^(x,y) = (\mu _{B_1}^\cup \mu _{B_2}^)(x,y)\\ \mu _{B_1+ B_2}^+(x,y) = (\mu _{B_1}^+\cup \mu _{B_2}^+)(x,y) \end{array}\right\}\) if \((x,y)\in E_1\cap E_2\),
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1+ B_2}^(x,y) = \min \{\mu _{A_1}^(x), \mu _{A_2}^(y)\}\\ \mu _{B_1+ B_2}^+(x,y) = \min \{\mu _{A_1}^+(x), \mu _{A_2}^+(y)\} \end{array}\right\}\) for all \((x,y)\in E'\), where \(E'\) is the set of edges connecting the vertices of \(V_1\) and \(V_2\).
Intervalvalued fuzzy \(\phi\)tolerance competition graph
In this section, the definition of intervalvalued fuzzy \(\phi\)tolerance competition graph is given and studied several properties.
Definition 1
Taking \(\phi\) as \(\min\). An example of this graph is given below.
Example 1
A \(\phi\)Tedge clique cover (\(\phi\)TECC) of an intervalvalued fuzzy graph \({\mathcal {G}}=(V,A,B)\) with vertices \(v_1,v_2,\ldots , v_n\) is a collection \(S_1,S_2,\ldots , S_k\) of subsets of V such that \(\mu _B^(v_r,v_s)>0\) if and only if at least \(\phi (c(T_r), c(T_s))\) of the sets \(S_i\), contain both \(v_r\) and \(v_s\). The size k of a smallest \(\phi\)TECC of \({\mathcal {G}}\) taken over all tolerances T is the \(\phi\)Tedge clique cover number and is denoted by \(\theta _{\phi }({\mathcal {G}})\).
Theorem 1
Let \(\phi {:}\,N\times N\rightarrow N\) be a mapping. If \(\theta _{\phi }({\mathcal {G}})\le V\), then there exists an intervalvalued fuzzy \(\phi\) tolerance competition graph.
Proof
Let us assume that \(\theta _{\phi }({\mathcal {G}})\le V\) and \(S_1,S_2,\ldots , S_k (k\le n)\) be a \(\phi\)TECC of an intervalvalued fuzzy graph \({\mathcal {G}}\). Each \(S_i\) is defined by \(S_i=\{v_j{:}\,\mu _B^(v_i, v_j)>0\}\). Each \(S_i\) is chosen in such a way that in the intervalvalued fuzzy digraph \(\overrightarrow{{\mathcal {G}}}=(V,A,\overrightarrow{B})\), \(\mu _B^(\overrightarrow{v_i,v_j})=\mu _{B'}^(v_i,v_j)\) and \(\mu _B^+(\overrightarrow{v_i,v_j})=\mu _{B'}^+(v_i,v_j)\), if \(v_j\in S_i\).
Now, in IVFG \({\mathcal {G}}\), either \(c({\mathcal {N}}^+(v_i)\cap {\mathcal {N}}^+(v_j))\ge \phi \{c({\mathcal {T}}_{v_i}), c({\mathcal {T}}_{v_j})\}\) or, \(s({\mathcal {N}}^+(v_i)\cap {\mathcal {N}}^+(v_j))\ge \phi \{s({\mathcal {T}}_{v_i}), s({\mathcal {T}}_{v_j})\}\) must satisfy.
Hence, \({\mathcal {G}}\) is an intervalvalued fuzzy \(\phi\)tolerance competition graph. \(\square\)
Theorem 2
For an intervalvalued fuzzy digraph \({\mathcal {G}}=(V,A,\overrightarrow{B})\), if there exists an intervalvalued fuzzy \(\phi\)tolerance competition graph, then \(\theta _{\phi }(\overrightarrow{{\mathcal {G}}})\le V=n.\)
Proof
Let \({\mathcal {G}}=(V,A,B')\) be an intervalvalued fuzzy \(\phi\)tolerance competition graph of \(\overrightarrow{G}\) and \(V=\{v_1,v_2,\ldots , v_n\}\) and \(S_i=\{v_j{:}\,\mu _{B'}^(v_i,v_j)>0\}\). It is clear that there can be at most n numbers of \(S_i\)’s.
Let \({\mathcal {T}}_1,{\mathcal {T}}_2,\ldots , {\mathcal {T}}_n\) be the fuzzy tolerances associated to each vertex of V.
Now, \(\mu (v_r,v_s)>0\) if and only if either \(c({\mathcal {N}}^+(v_r)\cap {\mathcal {N}}^+(v_s))\ge \phi \{c({\mathcal {T}}_{r}), c({\mathcal {T}}_{s})\}\) or, \(s({\mathcal {N}}^+(v_r)\cap {\mathcal {N}}^+(v_s))\ge \phi \{s({\mathcal {T}}_{r}), s({\mathcal {T}}_{s})\}\).
Thus, at most n sets \(S_1,S_2,\ldots , S_n\) make a family of \(\phi\)TECC of size at most \(n=V\), i.e. \(\theta _{\phi }(\overrightarrow{{\mathcal {G}}})\le V=n.\) \(\square\)
Theorem 3
Intervalvalued fuzzy \(\phi\)tolerance competition graph \(G=(V,A,B)\) cannot be complete.
Proof
Suppose, G be an intervalvalued fuzzy \(\phi\)tolerance competition graph with 2 vertices, x and y (say). For this graph there is no interval digraph with 2 vertices with some common preys. Hence, it cannot be complete.
If possible let, an IVFPTCG with 3 vertices be complete. Without any loss of generality, consider the graph of Fig. 3. This graph is nothing but a clique of order 3. As \(\mu _B(x,y)\ne [0,0]\), x, y has a common prey and it must be z. Thus, x, y is directed to z. Again \(\mu _B(y,z)\ne [0,0]\) implies that, y, z is directed to x. But in IVFDG, it is not possible to have two directed edges (x, z) and (z, x) simultaneously. This concludes that there is no valid IVFDG for this IVFPTCG.
As, every complete IVFPTCG contains a clique of order 3, there does not exist any valid IVFDG. Hence, any intervalvalued fuzzy \(\phi\)tolerance competition graph \(G=(V,A,B)\) cannot be complete. \(\square\)
Remark 1
The intervalvalued fuzzy \(\min\)tolerance competition graph of an irregular intervalvalued fuzzy digraph need not be irregular.
Consider the core and support lengths of fuzzy tolerances associated to each of the vertices of the irregular intervalvalued fuzzy digraph shown in Fig. 4 are 1, 1, 1 and 1, 1, 1 respectively.
Remark 2
The intervalvalued fuzzy \(\min\)tolerance competition graph of a regular intervalvalued fuzzy digraph need not be regular.
In Fig. 5, the regular intervalvalued fuzzy digraph has the degrees \(\deg (v_1)=\deg (v_2)=\cdots = \deg (v_5)=[0.7,0.9]\), but the degree of the vertices of intervalvalued fuzzy mintolerance competition graph of the digraph shown in Fig. 5 are \(\deg (v_1)=[0.4,0.5]\), \(\deg (v_2)=[0.6,0.8]\), \(\deg (v_3)=[0.2,0.3]\). Hence, it is not regular.
Definition 2
Theorem 4
Proof
Let \(ITC_{\phi }(\overrightarrow{{\mathcal {G}}})=(V,A,B')\) be the intervalvalued fuzzy \(\phi\)tolerance competition graph of an intervalvalued fuzzy digraph \(\overrightarrow{{\mathcal {G}}}=(V,A,\overrightarrow{B})\). As for every triangular orientation of three vertices in \(\overrightarrow{{\mathcal {G}}}\), as shown in Fig. 4, there is atmost one edge in \(ITC_{\phi }(\overrightarrow{{\mathcal {G}}})\), it is obvious that, an intervalvalued fuzzy \(\phi\)tolerance competition graph has less number of edges than that of the intervalvalued fuzzy digraph. Now, consider \(\mu _{B'}(v_1,v_2)>0\) in \(ITC_{\phi }(\overrightarrow{{\mathcal {G}}})\) and \({\mathcal {N}}^+(v_1)\) and \({\mathcal {N}}^+(v_2)\) has at least one vertex in common and also \(h({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_2))=[1,1]\) (as much as possible). Then there exist at least one vertex, say \(v_i\) so that the edge membership value between \(v_1\), \(v_i\) or \(v_2\), \(v_i\) is [1, 1]. Then \(S(\overrightarrow{{\mathcal {G}}})>[1,1]\) whereas, \(S(ITC_{\phi }(\overrightarrow{{\mathcal {G}}}))\le [1,1]\). Hence, \(S(ITC_{\phi }(\overrightarrow{{\mathcal {G}}}))\le S(\overrightarrow{{\mathcal {G}}}).\) \(\square\)
Theorem 5
If \(C_1,C_2,\ldots , C_p\) be the cliques of order 3 of underlying undirected crisp graph of a IVFDG \(\overrightarrow{G}=(V,A,\overrightarrow{B})\) such that \(C_1\cup C_2\cup \ldots C_p=V\) and \(C_i\cap C_j\le 1\) \(\forall i,j=1,2,\ldots , p\). Then the corresponding IVFPTCG of \(\overrightarrow{G}\) cannot have cliques of order 3 or more.
Proof
From the given conditions of clique sets, i.e. \(C_1\cup C_2\cup \ldots C_p=V\) and \(C_i\cap C_j\le 1 \forall i,j=1,2,\ldots , p\), it is clear that the intervalvalued fuzzy digraph has only triangular orientation and no two triangular orientation has a common edge. That is, the IVFDG has no orientation shown in Fig. 6b. The IVFDG only have the orientations of type shown in Fig. 6a.
As for every triangular orientation, there have only one edge in intervalvalued fuzzy \(\phi\)tolerance competition graph, the said graph does not have a clique of order 3 or more.
Hence, intervalvalued fuzzy \(\phi\)tolerance competition graph cannot have cliques of order 3 or more. \(\square\)
Theorem 6
If the clique number of an underlying undirected crisp graph of an intervalvalued fuzzy digraph \(\overrightarrow{{\mathcal {G}}}=(V,A,\overrightarrow{B})\) is p, then the underlying crisp graph of the intervalvalued fuzzy \(\phi\)tolerance competition graph has the clique number less than or equal to p.
Proof
Let us assume that the maximum clique of \(\overrightarrow{{\mathcal {G}}}=(V,A,\overrightarrow{B})\) induces a subgraph \(\overrightarrow{\mathcal {G'}}\) which is also an intervalvalued fuzzy directed graph. From Theorem 4, the size of intervalvalued fuzzy \(\phi\)tolerance competition graph is always less than or equal to the size of intervalvalued fuzzy directed graph, then the clique number of the intervalvalued fuzzy \(\phi\)tolerance competition graph cannot be greater than p. Hence the theorem follows.
Theorem 7
Intervalvalued fuzzy \(\phi\)tolerance competition graph of a complete intervalvalued fuzzy digraph has maximum \(^nC_3\) number of fuzzy edges.
Proof
It is obvious that every triangular orientation there exists an edge in IVFPTCG. Now, in a complete intervalvalued fuzzy digraph \(\mu _B^(x,y)=\min \{\mu _A^(x),\,\mu _A^(y)\}\), and \(\mu _B^+(x,y)=\min \{\mu _A^+(x),\mu _A^+(y)\}\), \(\forall x, y \in V\). Hence, every vertex is assigned to some vertex in V. Therefore, there are maximum \(^nC_3\) number of orientations. Therefore, there exists maximum \(^nC_3\) number of fuzzy edges in IVFPTCG. \(\square\)
Application of intervalvalued fuzzy maxtolerance competition graph in image matching
Product of two IVFPTCGs and relations between them
Throughout this paper, \(\theta\) is taken as the null set in crisp sense and \(\overrightarrow{G_1^*}\), \(\overrightarrow{G_2^*}\) are the crisp digraphs.
Definition 3
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1\times A_2}^(x_1, x_2) = \min \{\mu _{A_1}^(x_1), \mu _{A_2}^(x_2)\}\\ \mu ^+_{A_1\times A_2}(x_1, x_2) = \min \{\mu ^+_{A_1}(x_1), \mu ^+_{A_2}(x_2)\} \end{array}\right\}\) for all \(x_1\in V_1, x_2\in V_2\),
 (2)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^(\overrightarrow{(x,x_2),(x,y_2)}) = \min \{\mu _{A_1}^(x), \mu _{B_2}^(\overrightarrow{x_2,y_2})\}\\ \mu _{B_1\times B_2}^+(\overrightarrow{(x,x_2),(x,y_2)}) = \min \{\mu _{A_1}^+(x), \mu _{B_2}^+(\overrightarrow{x_2,y_2})\} \end{array}\right\}\) for all \(x\in V_1\) and \((\overrightarrow{x_2, y_2})\in E_2\),
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^(\overrightarrow{(x_1,y),(y_1,y)}) = \min \{\mu _{B_1}^(\overrightarrow{x_1,y_1}), \mu _{A_2}^(y)\}\\ \mu _{B_1\times B_2}^+(\overrightarrow{(x_1,y),(y_1,y)}) = \min \{\mu _{B_1}^+(\overrightarrow{x_1,y_1}), \mu _{A_2}^+(y)\} \end{array}\right\}\) for all \((\overrightarrow{x_1,y_1})\in E_1\) and \(y \in V_2\).
Theorem 8
Proof
It is easy to understand from the definition of IVFPTCG that all vertices and their membership values remain unchanged, but fuzzy edges and their membership values have been changed. Thus, there is no need to clarify about vertices.
Now, according to the definition of Cartesian product of two intervalvalued fuzzy directed graphs \(\overrightarrow{G_1}\) and \(\overrightarrow{G_2}\), there are two types of edges in \(\overrightarrow{G_1}\times \overrightarrow{G_2}\). The two cases are as follows.
Suppose, all edges are of type \(((x,x_2),(x,y_2))\), \(\forall x\in V_1\) and \((x_2,y_2)\in E_2\).
As, the either case is satisfied, therefore \(\mu _{B_1\times B_2}^((x,x_2),(x,y_2))>0\).
If all edges of type \(((x_1,y),(y_1,y))\), \(\forall y\in V_2\) and \((x_1,y_1)\in E_1\), then the proof is similar to above case.
Hence, \(ITC_{\phi }(\overrightarrow{G_1}\times \overrightarrow{G_2})= ITC_{\phi }(\overrightarrow{G_1})\times ITC_{\phi }(\overrightarrow{G_2})\) is proved. \(\square\)
Definition 4
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1\circ A_2}^(x_1, x_2) = \min \{\mu _{A_1}^(x_1), \mu _{A_2}^(x_2)\}\\ \mu ^+_{A_1\circ A_2}(x_1, x_2) = \min \{\mu ^+_{A_1}(x_1), \mu ^+_{A_2}(x_2)\} \end{array}\right\}\) for all \(x_1\in V_1, x_2\in V_2\),
 (2)
\(\left\{ \begin{array}{l} \mu _{B_1\circ B_2}^(\overrightarrow{(x,x_2),(x,y_2)}) = \min \{\mu _{A_1}^(x), \mu _{B_2}^(\overrightarrow{x_2,y_2})\}\\ \mu _{B_1\circ B_2}^+(\overrightarrow{(x,x_2),(x,y_2)}) = \min \{\mu _{A_1}^+(x), \mu _{B_2}^+(\overrightarrow{x_2,y_2})\} \end{array}\right\}\) for all \(x\in V_1\) and \((\overrightarrow{x_2, y_2})\in E_2\),
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1\circ B_2}^(\overrightarrow{(x_1,y),(y_1,y)}) = \min \{\mu _{B_1}^(\overrightarrow{x_1,y_1}), \mu _{A_2}^(y)\}\\ \mu _{B_1\circ B_2}^+(\overrightarrow{(x_1,y),(y_1,y)}) = \min \{\mu _{B_1}^+(\overrightarrow{x_1,y_1}), \mu _{A_2}^+(y)\} \end{array}\right\}\) for all \((\overrightarrow{x_1,y_1})\in E_1\) and \(y \in V_2\)
 (4)
\(\left\{ \begin{array}{l} \mu _{B_1\circ B_2}^(\overrightarrow{(x_1,x_2),(y_1,y_2)}) = \min \{\mu _{A_2}^(x_2), \mu _{A_2}^(y_2),\mu _{B_1}^(\overrightarrow{x_1,y_1})\}\\ \mu _{B_1\circ B_2}^+(\overrightarrow{(x_1,x_2),(y_1,y_2)}) = \min \{\mu _{A_2}^+(x_2), \mu _{A_2}^+(y_2),\mu _{B_1}(\overrightarrow{x_1,y_1})\} \end{array}\right\}\) otherwise.
Theorem 9
Proof
According to the same interpretation drawn in Theorem 8, the membership values of the vertices of \(\overrightarrow{G_1}[\overrightarrow{G_2}]\) remains unchanged under the composition \(\circ\).
Now, according to the definition of composition \(\overrightarrow{G_1}[\overrightarrow{G_2}]=(A_1\circ A_2, B_1\circ B_2)\) of two intervalvalued fuzzy directed graphs \(\overrightarrow{G_1}\) and \(\overrightarrow{G_2}\), there are three types of edges in \(\overrightarrow{G_1}\circ \overrightarrow{G_2}\). The three cases are as follows:
 Case I :

For all edges of type \(((x,x_2),(x,y_2))\), \(\forall x\in V_1\) and \((x_2,y_2)\in E_2\).
Obviously, from the definition of the Cartesian products of two directed graphs that, if \(x_2, y_2\) have a common prey \(z_2\) in \(\overrightarrow{G_2}\) then, \((x,x_2),(x,y_2)\) have also a common prey \((x,z_2)\) in \(\overrightarrow{G_1}\circ \overrightarrow{G_2}\), \(\forall x\in V_1\). Now, if \(\mu _{B_2}^(x_2,y_2)>0\) in \(ITC_{\phi }(\overrightarrow{G_2})\), then \(\mu _{B_1\circ B_2}^((x,x_2),(x,y_2))>0\) in \(ITC_{\phi }(\overrightarrow{G_1}\circ \overrightarrow{G_2})\). If \(\mu _{B_2}^(x_2,y_2)>0\), then either \(c({\mathcal {N}}^+(x_2)\cap {\mathcal {N}}^+(y_2))\ge \phi \{c({\mathcal {T}}_{x_2}), c({\mathcal {T}}_{y_2})\}\) or \(s({\mathcal {N}}^+(x_2)\cap {\mathcal {N}}^+(y_2))\ge \phi \{s({\mathcal {T}}_{x_2}), s({\mathcal {T}}_{y_2})\}\) is true. From the previous claim that if \(z_2\) is the common prey of \(x_2, y_2\) in \(\overrightarrow{G_2}\), \((x,z_2)\) is also a common prey of \((x,x_2)\) and \((x,y_2)\) in \(\overrightarrow{G_1}\circ \overrightarrow{G_2}\), then$$\begin{aligned} s({\mathcal {N}}^+(x,x_2)\cap {\mathcal {N}}^+(x,y_2))& = {} s({\mathcal {N}}^+(x_2)\cap {\mathcal {N}}^+(y_2))\\ &\ge \phi (s({\mathcal {T}}_{x_2}), s({\mathcal {T}}_{y_2}))\\ &\ge \phi (\min \{s({\mathcal {T}}_x),s( {\mathcal {T}}_{x_2})\},\min \{s({\mathcal {T}}_x),s({\mathcal {T}}_{y_2})\})\\& = {} \phi (s({\mathcal {T}}_{(x,x_2)}), s({\mathcal {T}}_{(x,y_2)})). \end{aligned}$$As, the either case is satisfied, \(\mu _{B_1\circ B_2}((x,x_2),(x,y_2))>0\) is true.
 Case II :

For all edges of type \(((x_1,y),(y_1,y))\), \(\forall y\in V_2\) and \((x_1,y_1)\in E_1\).
This is similar as the Case I.
 Case III :

For all edges of type \(((x_1,x_2),(y_1,y_2))\), where \(x_1\ne y_1\) and \(x_2\ne y_2\).
In this case, \((x_1,x_2)\) and \((y_1,y_2)\) have a common prey \((z_1,z_2)\) in \(\overrightarrow{G_1}\circ \overrightarrow{G_2}\) if \(x_1, y_1\) has a common prey \(z_1\) in \(\overrightarrow{G_1}\). In the similar way as in Case I, we can obtain$$\begin{aligned} s\left( {\mathcal {N}}^+(x_1,x_2)\cap {\mathcal {N}}^+(y_1,y_2)\right)& = {} s\left( {\mathcal {N}}^+(x_1)\cap {\mathcal {N}}^+(y_1)\right) \\ &\ge \phi \left( s({\mathcal {T}}_{x_1}), s({\mathcal {T}}_{y_1})\right) \\ &\ge \phi \left( \min \{s({\mathcal {T}}_{x_1}),s( {\mathcal {T}}_{x_2})\},\min \left\{ s({\mathcal {T}}_{y_1}),s({\mathcal {T}}_{y_2})\right\} \right) \\& = {} \phi \left( s({\mathcal {T}}_{(x_1,x_2)}), s({\mathcal {T}}_{(y_1,y_2)})\right) . \end{aligned}$$If, either case is satisfied, then \(\mu _{B_1\circ B_2}^((x_1,x_2),(y_1,y_2))>0\) is valid.
Hence, \(ITC_{\phi }(\overrightarrow{G_1}\circ \overrightarrow{G_2})= ITC_{\phi }(\overrightarrow{G_1})\circ ITC_{\phi }(\overrightarrow{G_2})\) is proved. \(\square\)
Definition 5
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1\cup A_2}^(x) =\mu _{A_1}^(x) {\text{if}}\,x\in V_1 {\hbox{and}} x\notin V_2\\ \mu _{A_1\cup A_2}^(x) =\mu _{A_2}^(x) {\text{if}}\,x\in V_2 {\hbox{and}} x\notin V_1\\ \mu _{A_1\cup A_2}^(x) =\max \{\mu _{A_1}^(x), \mu _{A_2}^(x)\} {\text{if}}\,x\in V_1\cap V_2. \end{array}\right.\)
 (2)
\(\left\{ \begin{array}{l} \mu _{A_1\cup A_2}^+(x) =\mu _{A_1}^+(x) {\text{if}}\,x\in V_1 {\hbox{and}} x\notin V_2\\ \mu _{A_1\cup A_2}^+(x) =\mu _{A_2}^+(x) {\text{if}}\,x\in V_2 {\hbox{and}} x\notin V_1\\ \mu _{A_1\cup A_2}^+(x) =\max \{\mu _{A_1}^+(x), \mu _{A_2}^+(x)\} {\text{if}}\,x\in V_1\cap V_2. \end{array}\right.\)
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^(\overrightarrow{x,y}) = \mu _{B_1}^(\overrightarrow{x,y}) {\text{if}}\,(\overrightarrow{x,y})\in E_1 {\text{and}}\,(\overrightarrow{x,y})\notin E_2\\ \mu _{B_1\times B_2}^(\overrightarrow{x,y}) = \mu _{B_2}^(\overrightarrow{x,y}) {\text{if}}\,(\overrightarrow{x,y})\in E_2 {\text{and}}\,(\overrightarrow{x,y})\notin E_1\\ \mu _{B_1\times B_2}^(\overrightarrow{x,y}) = \max \{\mu _{B_1}^(\overrightarrow{x,y}), \mu _{B_2}^(\overrightarrow{x,y})\} {\text{if}}\,(\overrightarrow{x,y})\in E_1\cap E_2. \end{array}\right.\)
 (4)
\(\left\{ \begin{array}{l} \mu _{B_1\times B_2}^+(\overrightarrow{x,y}) = \mu _{B_1}^+(\overrightarrow{x,y}) {\text{if}}\,(\overrightarrow{x,y})\in E_1 {\text{and}}\,(\overrightarrow{x,y})\notin E_2\\ \mu _{B_1\times B_2}^+(\overrightarrow{x,y}) = \mu _{B_2}^+(\overrightarrow{x,y}) {\text{if}}\,(\overrightarrow{x,y})\in E_2 {\text{and}}\,(\overrightarrow{x,y})\notin E_1\\ \mu _{B_1\times B_2}^+(\overrightarrow{x,y}) = \max \{\mu _{B_1}^+(\overrightarrow{x,y}), \mu _{B_2}^+(\overrightarrow{x,y})\} {\text{if}}\,(\overrightarrow{x,y})\in E_1\cap E_2. \end{array}\right.\)
Theorem 10
Proof
There are four cases as follows:
 Case I :

\(V_1\cap V_2=\theta\)
In this case, \(\overrightarrow{G_1}\cup \overrightarrow{G_2}\) is a disconnected intervalvalued fuzzy directed graphs with two components \(\overrightarrow{G_1}\) and \(\overrightarrow{G_2}\). Thus, there is nothing to prove that \(ITC_{\phi }(\overrightarrow{G_1}\cup \overrightarrow{G_2})= ITC_{\phi }(\overrightarrow{G_1})\cup ITC_{\phi }(\overrightarrow{G_2}).\)
 Case II :

\(V_1\cap V_2=\theta\), \((x_1,x_2)\in E_1\) and \((x_1,x_2)\notin E_2\)
\(\mu _{B_1\cup B_2}^(x_1,x_2)=\mu _{B_1}^(x_1,x_2)\) and it is obvious that if \(\mu _{B_1}^(x_1,x_2)>0\) in \(ITC_{\phi }(\overrightarrow{G_1})\), then \(\mu _{B_1\cup B_2}^(x_1,x_2)>0\) in \(ITC_{\phi }(\overrightarrow{G_1}\cup \overrightarrow{G_2})\).
 Case III :

\(V_1\cap V_2=\theta\), \((x_1,x_2)\notin E_1\) and \((x_1,x_2)\in E_2\)
This is similar as in Case II.
 Case IV :

\(V_1\cap V_2=\theta\), \((x_1,x_2)\in E_1\cap E_2\)
In this case, consider \(x_1\) and \(x_2\) have a common prey \(y_1\) in \(\overrightarrow{G_1}\) and \(y_2\) in \(\overrightarrow{G_2}\). This shows that \(s({\mathcal {N}}^+(x_1)\cap {\mathcal {N}}^+(x_2))\) in \(\overrightarrow{G_1}\cup \overrightarrow{G_2}\) is greater than or equal to \(s({\mathcal {N}}^+(x_1)\cap {\mathcal {N}}^+(x_2))\) in \(\overrightarrow{G_1}\) or \(\overrightarrow{G_2}\). Hence, it can be found that if \(\mu _{B_1}^(x_1,x_2)>0\) in \(ITC_{\phi }(\overrightarrow{G_1})\) and \(\mu _{B_2}^(x_1,x_2)>0\) in \(ITC_{\phi }(\overrightarrow{G_2})\), then \(\mu _{B_1\cup B_2}^(x_1,x_2)>0\) in \(ITC_{\phi }(\overrightarrow{G_1}\cup \overrightarrow{G_2})\).
Hence, \(ITC_{\phi }(\overrightarrow{G_1}\cup \overrightarrow{G_2})= ITC_{\phi }(\overrightarrow{G_1})\cup ITC_{\phi }(\overrightarrow{G_2})\) is proved. \(\square\)
Definition 6
 (1)
\(\left\{ \begin{array}{l} \mu _{A_1+ A_2}^(x) = (\mu _{A_1}^\cup \mu _{A_2}^)(x)\\ \mu _{A_1+ A_2}^+(x) = (\mu _{A_1}^+\cup \mu _{A_2}^+)(x) \end{array}\right\}\) if \(x\in V_1\cup V_2\),
 (2)
\(\left\{ \begin{array}{l} \mu _{B_1+ B_2}^(\overrightarrow{x,y}) = (\mu _{B_1}^\cup \mu _{B_2}^)(\overrightarrow{x,y})\\ \mu _{B_1+ B_2}^+(\overrightarrow{x,y}) = (\mu _{B_1}^+\cup \mu _{B_2}^+)(\overrightarrow{x,y}) \end{array}\right\}\) if \((\overrightarrow{x,y})\in E_1\cap E_2\),
 (3)
\(\left\{ \begin{array}{l} \mu _{B_1+ B_2}^(\overrightarrow{x,y}) = \min \{\mu _{A_1}^(x), \mu _{A_2}^(y)\}\\ \mu _{B_1+ B_2}^+(\overrightarrow{x,y}) = \min \{\mu _{A_1}^+(x), \mu _{A_2}^+(y)\} \end{array}\right\}\) for all \((\overrightarrow{x,y})\in E'\), where \(E'\) is the set of edges connecting the vertices (nodes) of \(V_1\) and \(V_2\).
Theorem 11
For any two intervalvalued fuzzy directed graphs \(\overrightarrow{G_1}\) and \(\overrightarrow{G_2}\), \(ITC_{\phi }(\overrightarrow{G_1}+ \overrightarrow{G_2})\) has less number of edges than that in \(ITC_{\phi }(\overrightarrow{G_1})+ ITC_{\phi }(\overrightarrow{G_2}).\)
Proof
In \(ITC_{\phi }(\overrightarrow{G_1})+ITC_{\phi }(\overrightarrow{G_2})\), \((\mu _{B_1}^+\mu _{B_2}^)(x_1,x_2)>0\) is true for all \(x_1\in V_1\) and \(x_2\in V_2\). But, in \(\overrightarrow{G_1}+\overrightarrow{G_2}\), \(x_1\) and \(x_2\) have no common prey, then \(\mu _{B_1+B_2}^(x_1,x_2)=0\) is valid for all \(x_1\in V_1\) and \(x_2\in V_2\). Thus, for all \(x_1, x_2\in V_1 \cup V_2\), \(\mu _{B_1+B_2}^(x_1,x_2)=0<(\mu _{B_1}^+\mu _{B_2}^)(x_1,x_2)\) is true always. Hence, the result follows. \(\square\)
Insights of this study

Intervalvalued fuzzy \(\phi\)tolerance competition graphs are introduced. The real life competitions in food web are perfectly represented by intervalvalued fuzzy \(\phi\)tolerance competition graphs.

An application of fuzzy \(\phi\)tolerance competition graph on image matching is provided. Particularly, intervalvalued fuzzy maxtolerance competition graph is used for this. Here, distorted images are matched for computer usages.

Product of two IVFPTCGs and relations between them are defined. These results will develop the theory of intervalvalued fuzzy graph literature. Some important results (Theorem 2, 3, 5, 9, 10) are proved.
Conclusions
Adding more uncertainty to fuzzy \(\phi\)tolerance competition graph, the intervalvalued fuzzy \(\phi\)tolerance competition graph was introduced here. Some interesting properties was investigated. Interesting properties of the IVFPTCG were proved such that the IVFPTCG of a IVFDG behaved like a homomorphic function under some operations. Generally, competition graphs represent some competitions in food webs. But, it can be also used in every competitive systems. These competitive systems can be represented by bipolar fuzzy graphs, intuitionistic fuzzy graphs, etc. But, interval valued fuzzy sets are perfect to represent uncertainties. An application of IVFPTCG in image matching was illustrated. Also, it can be applied in various types of fields such as database management system, network designing, neural network, image searching in computer application, etc.
Notes
Declarations
Authors' contributions
The authors contributed equally to each parts of the paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the Editor in Chief and Honorable reviewers of the journal “Springer Plus” for their suggestions to improve the quality and presentation of the paper.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Akram M, Dudek WA (2011) Interval valued fuzzy graphs. Comput Math Appl 61:289–299MathSciNetView ArticleMATHGoogle Scholar
 Bhutani KR, Battou A (2003) On Mstrong fuzzy graphs. Inf Sci 155(1–2):103–109MathSciNetView ArticleMATHGoogle Scholar
 Bhutani KR, Rosenfeld A (2003) Strong arcs in fuzzy graphs. Inf Sci 152:319–322MathSciNetView ArticleMATHGoogle Scholar
 Brigham RC, McMorris FR, Vitray RP (1995) Tolerance competition graphs. Linear Algebra Appl 217:41–52MathSciNetView ArticleMATHGoogle Scholar
 Cho HH, Kim SR, Nam Y (2000) The \(m\)step competition graph of a digraph. Discrete Appl Math 105:115–127MathSciNetView ArticleMATHGoogle Scholar
 Cohen JE (1968) Interval graphs and food webs: a finding and a problem, Document 17696PR. RAND Corporation, Santa MonicaGoogle Scholar
 Golumbic MC, Monma CL (1982) A generalization of interval graphs with tolerances. In: Proceedings of the 13th Southeastern conference on combinatories, graph theory and computing, Congressus Numerantium Utilitas Math, Winnipeg, pp 321–331Google Scholar
 Kauffman A (1973) Introduction a la Theorie des Sousemsembles Flous. Masson et Cie Editeurs, ParisGoogle Scholar
 Kim SR, McKee TA, McMorris FR, Roberts FS (1995) \(p\)Competition graphs. Discrete Appl Math 217:167–178MathSciNetMATHGoogle Scholar
 Koczy LT (1992) Fuzzy graphs in the evaluation and optimization of networks. Fuzzy Sets Syst 46:307–319MathSciNetView ArticleMATHGoogle Scholar
 Mathew S, Sunitha MS (2009) Types of arcs in a fuzzy graph. Inf Sci 179:1760–1768MathSciNetView ArticleMATHGoogle Scholar
 Mordeson JN, Nair PS (2000) Fuzzy graphs and fuzzy hypergraphs. Physica, HeidelbergView ArticleMATHGoogle Scholar
 Pramanik T, Samanta S, Pal M (2014) Intervalvalued fuzzy planar graphs. Int J Mach Learn Cybern. doi:https://doi.org/10.1007/s1304201402847 Google Scholar
 Pramanik T, Samanta S, Sarkar B, Pal M (2016) Fuzzy phitolerance competition graphs. Soft Comput. doi:https://doi.org/10.1007/s0050001520265 Google Scholar
 Samanta S, Pal M (2015) Fuzzy planar graphs. IEEE Trans Fuzzy Syst 23(6):1936–1942View ArticleGoogle Scholar
 Samanta S, Pal M, Akram M (2014) \(m\)step fuzzy competition graphs. J Appl Math Comput. doi:https://doi.org/10.1007/s1219001407852 MathSciNetMATHGoogle Scholar
 Samanta S, Pal M (2013) Fuzzy \(k\)competition graphs and \(p\)competition fuzzy graphs. Fuzzy Eng Inf 5(2):191–204MathSciNetView ArticleGoogle Scholar
 Samanta S, Pal M (2011) Fuzzy tolerance graphs. Int J Latest Trends Math 1(2):57–67Google Scholar
 Rosenfeld A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications. Academic Press, New York, pp 77–95Google Scholar