In this section, the definition of interval-valued fuzzy \(\phi\)-tolerance competition graph is given and studied several properties.

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**Definition 1**

(*Interval-valued fuzzy*
\(\phi\)-*tolerance competition graph* (*IVFPTCG*)) Let \(\phi {:}\,N\times N\rightarrow N\) be a mapping, where *N* is a set of natural numbers. Interval-valued fuzzy \(\phi\)-tolerance competition graph of an interval-valued fuzzy directed graph (IVFDG) \(\overrightarrow{D}=(V,A,\overrightarrow{B})\) is an undirected graph \(ITC_{\phi }(\overrightarrow{D}) = (V,A, B')\) such that

$$\begin{aligned} \mu _{B'} (u,v) &= {} [\mu _{B'}^-(u,v), \mu _{B'}^+(u,v)]\\& = {} \left\{ \begin{array}{l} h({{\mathcal {N}}}^+(u)\cap {\mathcal {N}}^+(v)),\\ \,\quad \qquad \text{ if } c({\mathcal {N}}^+(u)\cap {\mathcal {N}}^+(v))\ge \phi \{c({\mathcal {T}}_u), c({\mathcal {T}}_v)\}\\ \frac{s({\mathcal {N}}^+(u)\cap {\mathcal {N}}^+(v))-\phi \{s({\mathcal {T}}_u), s({\mathcal {T}}_v))\}+1}{s({\mathcal {N}}^+(u)\cap {\mathcal {N}}^+(v))}\cdot h({\mathcal {N}}^+(u)\cap {\mathcal {N}}^+(v)),\\ \,\quad \qquad \text{ if } s({\mathcal {N}}^+(u)\cap {\mathcal {N}}^+(v))\ge \phi \{s({\mathcal {T}}_u), s({\mathcal {T}}_v)\}\\ 0, \,\quad \text{ otherwise. } \end{array} \right. \end{aligned}$$

where, \({\mathcal {T}}_u, {\mathcal {T}}_v\) are the fuzzy tolerances corresponding to *u* and *v*, respectively.

Taking \(\phi\) as \(\min\). An example of this graph is given below.

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*Example 1*

Consider an interval-valued fuzzy digraph \(\overrightarrow{G}=(V,A,\overrightarrow{B})\) shown in Fig. 2 with each vertex have membership values [1, 1]. The edge membership values are taken as

$$\begin{aligned} &\mu _B(\overrightarrow{v_1,v_2})=[0.8,0.9], \quad \mu _B(\overrightarrow{v_1,v_5})=[0.7,0.8],\\ &\mu _B(\overrightarrow{v_2,v_5})=[0.6,0.8], \quad \mu _B(\overrightarrow{v_3,v_2})=[0.5,0.7],\\ &\mu _B(\overrightarrow{v_3,v_4})=[0.3,0.5], \quad \mu _B(\overrightarrow{v_4,v_1})=[0.7,0.9],\\ &\mu _B(\overrightarrow{v_5,v_3})=[0.6,0.8],\quad \mu _B(\overrightarrow{v_5,v_4})=[0.5,0.6]. \end{aligned}$$

Let core and support lengths of fuzzy tolerances \({\mathcal {T}}_1,{\mathcal {T}}_2, {\mathcal {T}}_3,{\mathcal {T}}_4,{\mathcal {T}}_5\) corresponding to the vertices \(v_1, v_2,v_3,v_4,v_5\) be 1, 1, 3, 2, 0 and 1, 2, 4, 3, 1, respectively. Here, it is true that \(\phi \{c({\mathcal {T}}_u), c({\mathcal {T}}_v)\}=\min \{c({\mathcal {T}}_u), c({\mathcal {T}}_v)\}\).

Based on this consideration, the following computations have been made.

$$\begin{aligned} {\mathcal {N}}^+(v_1)& = {} \{(v_2,[0.8,0.9]),(v_5,[0.7,0.8])\}\\ {\mathcal {N}}^+(v_2)& = {} \{(v_5,[0.6,0.8])\}\\ {\mathcal {N}}^+(v_3)& = {} \{(v_2,[0.5,0.7]),(v_4,[0.3,0.5])\}\\ {\mathcal {N}}^+(v_4)& = {} \{(v_1,[0.7,0.9])\}\\ {\mathcal {N}}^+(v_5)& = {} \{(v_3,[0.6,0.8]),(v_4,[0.5,0.6])\} \end{aligned}$$

Therefore,

$$\begin{aligned}&{\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_2)=\{(v_5,[0.6,0.8])\}\\&{\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_3)=\{(v_2,[0.5,0.7])\}\\&{\mathcal {N}}^+(v_3)\cap {\mathcal {N}}^+(v_5)=\{(v_4,[0.3,0.5])\} \end{aligned}$$

Then

$$\begin{aligned}&h({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_2))=[0.6,0.8]\\&h({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_3))=[0.5,0.7]\\&h({\mathcal {N}}^+(v_3)\cap {\mathcal {N}}^+(v_5))=[0.3,0.5] \end{aligned}$$

Now,

$$\begin{aligned}&c({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_2))=0; s({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_2))=1\\&c({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_3))=0; s({\mathcal {N}}^+(v_1)\cap {\mathcal {N}}^+(v_3))=1\\&c({\mathcal {N}}^+(v_3)\cap {\mathcal {N}}^+(v_5))=0; s({\mathcal {N}}^+(v_3)\cap {\mathcal {N}}^+(v_5))=1. \end{aligned}$$

Then by the definition of interval-valued fuzzy \(\phi\)-tolerance competition graph, the vertex membership function of the interval-valued fuzzy min-tolerance competition graph is that of interval-valued fuzzy digraph shown in Fig. 2 and the edge membership values are as follows:

$$\begin{aligned} \begin{array}{ll} \mu _B({v_1,v_3})=[0.5,0.7], &{}\quad \mu _B({v_1,v_2})=[0.6,0.8],\\ \mu _B({v_3,v_5})=[0.3,0.5]. \end{array} \end{aligned}$$

A \(\phi\)-T-edge clique cover (\(\phi\)-T-ECC) of an interval-valued 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 \(\phi\)-T-ECC of \({\mathcal {G}}\) taken over all tolerances *T* is the \(\phi\)-T-edge clique cover number and is denoted by \(\theta _{\phi }({\mathcal {G}})\).

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**Theorem 1**

*Let*
\(\phi {:}\,N\times N\rightarrow N\)
*be a mapping. If*
\(\theta _{\phi }({\mathcal {G}})\le |V|\), *then there exists an interval-valued 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 \(\phi\)-T-ECC of an interval-valued 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 interval-valued 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 interval-valued fuzzy \(\phi\)-tolerance competition graph. \(\square\)

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**Theorem 2**

*For an interval-valued fuzzy digraph*
\({\mathcal {G}}=(V,A,\overrightarrow{B})\), *if there exists an interval-valued fuzzy*
\(\phi\)-*tolerance competition graph, then*
\(\theta _{\phi }(\overrightarrow{{\mathcal {G}}})\le |V|=n.\)

###
*Proof*

Let \({\mathcal {G}}=(V,A,B')\) be an interval-valued 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 \(\phi\)-T-ECC of size at most \(n=|V|\), i.e. \(\theta _{\phi }(\overrightarrow{{\mathcal {G}}})\le |V|=n.\)
\(\square\)

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**Theorem 3**

*Interval-valued fuzzy*
\(\phi\)-*tolerance competition graph*
\(G=(V,A,B)\)
*cannot be complete.*

###
*Proof*

Suppose, *G* be an interval-valued 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 interval-valued fuzzy \(\phi\)-tolerance competition graph \(G=(V,A,B)\) cannot be complete. \(\square\)

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*Remark 1*

The interval-valued fuzzy \(\min\)-tolerance competition graph of an irregular interval-valued fuzzy digraph need not be irregular.

This can be shown by giving a counter-example. Suppose an interval-valued fuzzy digraph with 3 vertices shown in Fig. 4.

Consider the core and support lengths of fuzzy tolerances associated to each of the vertices of the irregular interval-valued fuzzy digraph shown in Fig. 4 are 1, 1, 1 and 1, 1, 1 respectively.

###
*Remark 2*

The interval-valued fuzzy \(\min\)-tolerance competition graph of a regular interval-valued fuzzy digraph need not be regular.

To prove this, a counter-example is given in the Fig. 5.

In Fig. 5, the regular interval-valued 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 interval-valued fuzzy min-tolerance 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.

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**Definition 2**

The *size* of an interval-valued fuzzy graph \({\mathcal {G}}=(V,A, B)\) is denoted by \(S({\mathcal {G}})\) and is defined by

$$\begin{aligned} S({\mathcal {G}})= \sum \mu _B(u,v)=\left[ \sum \mu _B^-(u,v), \sum \mu _B^+(u,v)\right] . \end{aligned}$$

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**Theorem 4**

*Let*
\(\overrightarrow{{\mathcal {G}}}\)
*be an interval-valued fuzzy digraph and*
\(ITC_{\phi }(\overrightarrow{{\mathcal {G}}})\)
*be its interval-valued fuzzy*
\(\phi\)
*-tolerance competition graph. Then*

$$\begin{aligned} S(ITC_{\phi }(\overrightarrow{{\mathcal {G}}}))\le S(\overrightarrow{{\mathcal {G}}}). \end{aligned}$$

###
*Proof*

Let \(ITC_{\phi }(\overrightarrow{{\mathcal {G}}})=(V,A,B')\) be the interval-valued fuzzy \(\phi\)-tolerance competition graph of an interval-valued 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 interval-valued fuzzy \(\phi\)-tolerance competition graph has less number of edges than that of the interval-valued 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, \(S(ITC_{\phi }(\overrightarrow{{\mathcal {G}}}))\le S(\overrightarrow{{\mathcal {G}}}).\)
\(\square\)

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**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.*

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*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 interval-valued 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 interval-valued fuzzy \(\phi\)-tolerance competition graph, the said graph does not have a clique of order 3 or more.

Hence, interval-valued fuzzy \(\phi\)-tolerance competition graph cannot have cliques of order 3 or more. \(\square\)

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**Theorem 6**

*If the clique number of an underlying undirected crisp graph of an interval-valued fuzzy digraph*
\(\overrightarrow{{\mathcal {G}}}=(V,A,\overrightarrow{B})\)
*is*
*p*, *then the underlying crisp graph of the interval-valued 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 interval-valued fuzzy directed graph. From Theorem 4, the size of interval-valued fuzzy \(\phi\)-tolerance competition graph is always less than or equal to the size of interval-valued fuzzy directed graph, then the clique number of the interval-valued fuzzy \(\phi\)-tolerance competition graph cannot be greater than *p*. Hence the theorem follows.

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**Theorem 7**

*Interval-valued fuzzy*
\(\phi\)-*tolerance competition graph of a complete interval-valued 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 interval-valued 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. \(\square\)