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Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 91–127 | Cite as

Rough fuzzy digraphs with application

  • Muhammad AkramEmail author
  • Fariha Zafar
Original Research

Abstract

Rough set theory is a mathematical tool to deal with incomplete and vague information. Fuzzy set theory deals the problem of how to understand and manipulate imperfect knowledge. The aim of this research is to construct a framework for handling vague information by applying some new concept of rough fuzzy digraphs. In this research study, we present certain new aspects of rough fuzzy digraphs (RFDs) based on rough fuzzy set model. We discuss complement and \(\mu \)-complement of RFDs. We discuss the concept of isomorphisms between RFDs and the irregularity of RFDs in detail. We consider an application of our proposed hybrid decision-making method: RFDs. We also describe our hybrid decision-making method as an algorithm.

Keywords

Rough fuzzy relation Irregular rough fuzzy digraphs Hybrid decision-making method Algorithm 

Mathematics Subject Classification

03E72 68R10 68R05 

1 Introduction

Due to recent advances in science and technology, traditional mathematical tools are not sufficient for dealing with the complex problems arising in our real world day by day. To address these increasing challenges, there is need of novel and innovative mathematical tools. A biggest dilemma of our universe is uncertainty and traditional crisp methods fail to handle these uncertainties in some complex problems. Many researchers extended the classical sets to various new models like fuzzy sets [26], soft sets [17], intuitionistic fuzzy sets [5], interval-valued fuzzy sets [24], vague sets [15], rough sets [21] and many others to address the problems related to vagueness and uncertainty. Rough set theory (RST) was developed by Pawlak [21] in 1982, which is an analyzing technique for managing the imperfection and imprecise data. Any subset of objects of a universe is approximated by two sets of approximations which are called the lower approximation (TLA) and the upper approximation (TUA). RST incorporates with the uncertainty caused by the lack of discernible objects.

Due to the limitation of human’s knowledge to understand the complex problems, it is very difficult to apply only a single type of uncertainty method to deal with such problems. Therefore, it is necessary to develop hybrid models by incorporating the advantages of many other different mathematical models dealing uncertainty. Many scholars Dubois and Prade [10], Pawlak [22], Biswas [7, 8], Banerjee and Pal [6] and Chakrabarty et al. [9] considered different aspects of objects, universe and relations and built hybrid models of fuzzy and rough sets, which are different from each other but are related to some extent. These models are rough fuzzy sets (RFSs), fuzzy rough sets (FRSs) and generalized fuzzy rough sets (GFRSs). FRS approximates a fuzzy set under a fuzzy environment, that is, in fuzzy rough set theory, a fuzzy equivalence relation on the given set of objects is considered which makes a fuzzy approximation space and a fuzzy set on the set of objects is characterized w.r.t the fuzzy approximation space. Whereas, RFS approximates a fuzzy set under a crisp environment, that is, in rough fuzzy set theory, a crisp equivalence relation on the given set of objects is considered which makes a crisp approximation space and then a fuzzy set on the set of objects is characterized w.r.t the crisp approximation space. In generalized fuzzy rough set, the condition of fuzzy equivalence relation is replaced by any arbitrary fuzzy relation. In 1996, Pawlak [23] introduced rough relations. Rough sets have been successfully applied in the fields of attribute reduction, decision-making, feature selection and rule extraction. Moreover, many new rough set models have also been established by combining the Pawlak rough set with other uncertainty theories such as soft set theory. Feng et al. [11, 12, 13, 14] provided a framework to combine fuzzy sets, rough sets, and soft sets all together, which gives rise to some new hybrid models such as rough soft sets, soft rough sets, and soft rough fuzzy sets.

In 1973, fuzzy graphs were introduced by Kauffman [16]. Some operations on fuzzy graphs were developed by Mordeson and Peng [18]. Nagoorgani [19] discussed the properties of \(\mu \)-complement of a fuzzy graph. Nagoorgani and Latha [20] discussed the irregularity of fuzzy graphs. Bipolar fuzzy graphs were first proposed by Akram [1]. A simple graph having directed edges is called a digraph. Arrows on the edges encode a directional information, i.e., an arc from a vertex w to another vertex z denotes that one can go from w to z but not from z to w. Wu [25] introduced fuzzy digraphs (FDs) in 1986. Akram et al. [2] presented novel applications of intuitionistic FDs in decision support systems. Zafar and Akram [27] considered a novel DM method based on a hybrid model RFDs. In this research study, we present certain new aspects of rough fuzzy digraphs (RFDs). We discuss some new operations on RFDs and the complement and \(\mu \)-complement of RFDs. We discuss the certain types of irregularity of RFDs in detail and describe their properties. We discuss the concept of isomorphisms between RFDs. We consider an application of our proposed hybrid decision-making method: RFDs. We also describe our hybrid decision-making method as an algorithm. For other notions and definitions, the readers are referred to [3, 4, 28, 29, 30].

2 New aspects of rough fuzzy digraphs

Definition 2.1

[10] Let Z be a universe and M an equivalence relation (ER) on Z. Let \(J\in {\mathcal {F}}(Z)\), where \({\mathcal {F}}(Z)\) represents the fuzzy power set. The TLA and TUA of the fuzzy set J, represented by \(\underline{M}J\) and \(\overline{M}J\), respectively, are characterized by fuzzy sets in Z such that, for all \(w\in Z\),
$$\begin{aligned} (\underline{M}J)(w)= & {} \bigwedge \limits _{w_1\in Z}\big ((1-M(w,w_1))\vee J(w_1)\big ), \\ (\overline{M}J)(w)= & {} \bigvee \limits _{w_1\in Z}\big (M(w,w_1)\wedge J(w_1)\big ). \end{aligned}$$
The pair \(MJ=(\underline{M}J,\overline{M}J)\) is called a rough fuzzy set.

Definition 2.2

[27] Let \(J^*\) be a nonempty set and M an ER on \(J^*\). Let J be a fuzzy set on \(J^*\) and \(MJ=(\underline{M}J,\overline{M}J)\) a RFS. Let \(K^*\subseteq J^*\times J^*\). Let N be an ER on \(K^*\) such that
$$\begin{aligned} (wz,w_{1}z_{1})\in N \Longleftrightarrow (w,w_{1}), (z,z_{1})\in M,~\forall ~wz, w_{1}z_{1}\in K^*. \end{aligned}$$
Let K be a fuzzy set on \(K^*\subseteq J^*\times J^*\) such that
$$\begin{aligned} K(wz)\le \min \{(\underline{M}J)(w),(\underline{M}J)(z)\},~~\forall ~w,z\in J^*. \end{aligned}$$
Then the TLA and TUA of K, represented by \(\underline{N}K\) and \(\overline{N}K\), respectively, are characterized as fuzzy sets in \(J^*\times J^*\) such that, \(\forall ~ wz\in K^*\),
$$\begin{aligned} (\underline{N}K)(wz)= & {} \bigwedge \limits _{w_1z_1\in K^*}\big ((1-N(wz,w_1z_1))\vee K(w_1z_1)\big ),\\ (\overline{N}K)(wz)= & {} \bigvee \limits _{w_1z_1\in K^*}\big (N(wz, w_1z_1)\wedge K(w_1z_1)\big ). \end{aligned}$$
The pair \(NK=(\underline{N}K,\overline{N}K)\) is called a rough fuzzy relation(RFR).

Definition 2.3

[27] A rough fuzzy digraph on a nonempty set \(J^*\) is an 4-ordered tuple \(\hat{G}=(M,MJ,N,NK)\) such that
  1. (a)

    M is an ER on \(J^*\),

     
  2. (b)

    N is an ER on \(K^*\subseteq J^*\times J^*\),

     
  3. (c)

    \(MJ=(\underline{M}J,\overline{M}J))\) is a RFS on \(J^*\),

     
  4. (d)

    \(NK=(\underline{N}K,\overline{N}K)\) is a RFR on \(J^*\),

     
  5. (e)
    (MJNK) is a fuzzy digraph, where \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are lower and upper approximate fuzzy digraphs(FDs) of \(\hat{G}\) such that
    $$\begin{aligned} (\underline{N}K)(wz)\le & {} \min \{(\underline{M}J)(w),(\underline{M}J)(z)\}, \\ (\overline{N}K)(wz)\le & {} \min \{(\overline{M}J)(w),(\overline{M}J)(z)\}, ~~~\forall ~w,z\in J^*. \end{aligned}$$
     

Example 2.1

Let \(J^*=\{t_1,t_2,t_3,t_4,t_5\}\) be a set and M an ER on \(J^*\) defined by
$$\begin{aligned} \begin{array}{c|@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} M &{} t_1 &{} t_2 &{} t_3 &{} t_4 &{} t_5 \\ \hline t_1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ t_2 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ t_3 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ t_4 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ t_5 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1\\ \end{array} \end{aligned}$$
Let \(J=\{(t_1,0.8),(t_2,0.6),(t_3,0.7),(t_4,0.4),(t_5,0.9)\}\) be a fuzzy set on \(J^*\) and \(MJ=(\underline{M}J,\overline{M}J)\) a RFS, where \(\underline{M}J\) and \(\overline{M}J\) are defined by
$$\begin{aligned} \underline{M}J= & {} \{(t_1,0.7),(t_2,0.6),(t_3,0.7),(t_4,0.4),(t_5,0.4)\}, \\ \overline{M}J= & {} \{(t_1,0.8),(t_2,0.6),(t_3,0.8),(t_4,0.9),(t_5,0.9)\}. \end{aligned}$$
Let \(K^*=\{t_1t_1,t_1t_3,t_1t_4,t_2t_1,t_2t_5,t_3t_5,t_4t_2,t_4t_3,t_4t_4,t_5t_4\}\subseteq J^*\times J^*\) and N an ER on \(K^*\) defined by
$$\begin{aligned} \begin{array}{c|@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} N &{} t_1t_1 &{} t_1t_3 &{} t_1t_4 &{} t_2t_1 &{} t_2t_5 &{} t_3t_5 &{} t_4t_2 &{} t_4t_3 &{} t_4t_4 &{} t_5t_4 \\ \hline t_1t_1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_1t_3 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_1t_4 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_2t_1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_2t_5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_3t_5 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_4t_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ t_4t_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ t_4t_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ t_5t_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1\\ \end{array} \end{aligned}$$
Let \(K=\{(t_1t_1,0.6),(t_1t_3,0.6),(t_1t_4,0.3),(t_2t_1,0.5),(t_2t_5,0.4),(t_3t_5,0.2),(t_4t_2,0.2),(t_4t_3,0.4), (t_4t_4,0.3),(t_5t_4,0.1)\}\) be a fuzzy set on \(K^*\) and \(NK=(\underline{N}K,\overline{N}K)\) a RFR, where \(\underline{N}K\) and \(\overline{N}K\) are defined by
$$\begin{aligned} \underline{N}K= & {} \{(t_1t_1,0.6),(t_1t_3,0.6),(t_1t_4,0.2),(t_2t_1,0.5),(t_2t_5,0.4),(t_3t_5,0.2),(t_4t_2,0.2),\\&~ (t_4t_3,0.4),(t_4t_4,0.1),(t_5t_4,0.1)\}, \\ \overline{N}K= & {} \{(t_1t_1,0.7),(t_1t_3,0.7),(t_1t_4,0.3),(t_2t_1,0.5),(t_2t_5,0.4),(t_3t_5,0.3),(t_4t_2,0.2),\\&~ (t_4t_3,0.4),(t_4t_4,0.3),(t_5t_4,0.3)\}. \end{aligned}$$
Thus, \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are FDs as shown in Fig. 1.
Hence, \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\) is a RFD.
Fig. 1

Rough fuzzy digraph \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\)

Definition 2.4

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). The order of \(\hat{G}\), denoted by \({\mathbb {O}}(\hat{G})\), represented by
$$\begin{aligned} {\mathbb {O}}(\hat{G})={{\mathbb {O}}}(\underline{\hat{G}})+{\mathbb {O}}(\overline{\hat{G}}), \end{aligned}$$
where
$$\begin{aligned} {\mathbb {O}}(\underline{\hat{G}})= & {} \sum \limits _{w\in J^*}(\underline{M}J)(w),\\ {\mathbb {O}}(\overline{\hat{G}})= & {} \sum \limits _{w\in J^*}(\overline{M}J)(w). \end{aligned}$$
The size of \(\hat{G}\), denoted by \({\mathbb {S}}(\hat{G})\), represented by
$$\begin{aligned} {\mathbb {S}}(\hat{G})={\mathbb {S}}(\underline{\hat{G}})+{\mathbb {S}}(\overline{\hat{G}}), \end{aligned}$$
where
$$\begin{aligned} {\mathbb {S}}(\underline{\hat{G}})= & {} \sum \limits _{w,z\in J^*}(\underline{N}K)(wz),\\ {\mathbb {S}}(\overline{\hat{G}})= & {} \sum \limits _{w,z\in J^*}(\overline{N}K)(wz). \end{aligned}$$

Example 2.2

Let \(\hat{G}\) be a RFD as shown in Fig. 1. Then
$$\begin{aligned} {\mathbb {O}}(\underline{\hat{G}})= & {} 0.7+0.6+0.7+0.4+0.4= 2.8,\\ {\mathbb {O}}(\overline{\hat{G}})= & {} 0.8+0.6+0.8+0.9+0.9= 4.0,\\ {\mathbb {O}}(\hat{G})= & {} 2.8+4.0 = 6.8. \end{aligned}$$
and
$$\begin{aligned} {\mathbb {S}}(\underline{\hat{G}})= & {} 0.6+0.6+0.2+0.5+0.4+0.2+0.2+0.4+0.1+0.1 = 3.3,\\ {\mathbb {S}}(\overline{\hat{G}})= & {} 0.7+0.7+0.3+0.5+0.4+0.3+0.2+0.4+0.3+0.3 = 4.1,\\ {\mathbb {S}}(\hat{G})= & {} 3.3+4.1 = 7.4. \end{aligned}$$

Definition 2.5

The underlying crisp digraph of a RFD \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\) denoted by \(\hat{G}^*\), represented by \(\hat{G}^*=\underline{\hat{G}}^*\) or \(\hat{G}^* =\overline{\hat{G}}^*\) such that
$$\begin{aligned} (\underline{M}J)^*= & {} \{w\in J^*|(\underline{M}J)(w)> 0\}, \\ (\overline{M}J)^*= & {} \{w\in J^*|(\overline{M}J)(w)> 0\}, \\ (\underline{N}K)^*= & {} \{wz\in K^*|(\underline{N}K)(wz)>0\},\\ (\overline{N}K)^*= & {} \{wz\in K^*|(\overline{N}K)(wz)> 0\}. \end{aligned}$$

Example 2.3

Consider the RFD \(\hat{G}\) as shown in Fig. 1. The underlying crisp digraph of \(\hat{G}\) is \(\hat{G}^* =\underline{\hat{G}}^*=\overline{\hat{G}}^*\) and shown in Fig. 2.

Fig. 2

Underlying crisp digraph \(\hat{G}^*\) of \(\hat{G}\)

Definition 2.6

Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*\). The union of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a RFD \(\hat{G}=\hat{G}_{1}\Cup \hat{G}_{2}=(\underline{\hat{G}}_{1}\cup \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\cup \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\cup \underline{\hat{G}}_{2}=(\underline{M}J_{1}\cup \underline{M}J_{2}, \underline{N}K_{1}\cup \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\cup \overline{\hat{G}}_{2}=(\overline{M}J_{1}\cup \overline{M}J_{2}, \overline{N}K_{1}\cup \overline{N}K_{2})\) are FDs, respectively, such that
$$\begin{aligned}&(\underline{M}J_{1}\cup \underline{M}J_{2})(w)\\&\quad = \left\{ \begin{array}{ll} (\underline{M}J_{1})(w), &{} \hbox {if }w\in (\underline{M}J_{1})^* \hbox { but }w\not \in (\underline{M}J_{2})^*; \\ (\underline{M}J_{2})(w), &{} \hbox {if }w\in (\underline{M}J_{2})^* \hbox { but }w\not \in (\underline{M}J_{1})^*; \\ \max \{(\underline{M}J_{1})(w), (\underline{M}J_{2})(w)\}, &{} \hbox {if } w\in (\underline{M}J_{1})^*\cap (\underline{M}J_{2})^*. \end{array} \right. \\&(\underline{N}K_{1}\cup \underline{N}K_{2})(wz)\\&\quad = \left\{ \begin{array}{ll} (\underline{N}K_{1})(wz), &{} \hbox {if }wz\in (\underline{N}K_{1})^* \hbox { but }wz\not \in (\underline{N}K_{2})^*; \\ (\underline{N}K_{2})(wz), &{} \hbox {if }wz\in (\underline{N}K_{2})^* \hbox { but }wz\not \in (\underline{N}K_{1})^*; \\ \max \{(\underline{N}K_{1})(wz), (\underline{N}K_{2})(wz)\}, &{} \hbox {if }wz\in (\underline{N}K_{1})^*\cap (\underline{N}K_{2})^*. \end{array} \right. \\&(\overline{M}J_{1}\cup \overline{M}J_{2})(w)\\&\quad = \left\{ \begin{array}{ll} (\overline{M}J_{1})(w), &{} \hbox {if }w\in (\overline{M}J_{1})^*\hbox { but } w\not \in (\overline{M}J_{2})^*; \\ (\overline{M}J_{2})(w), &{} \hbox {if }w\in (\overline{M}J_{2})^*\hbox { but } w\not \in (\overline{M}J_{1})^*; \\ \max \{(\overline{M}J_{1})(w), (\overline{M}J_{2})(w)\}, &{} \hbox {if } w\in (\overline{M}J_{1})^*\cap (\overline{M}J_{2})^*. \end{array} \right. \\&(\overline{N}K_{1}\cup \overline{N}K_{2})(wz)\\&\quad = \left\{ \begin{array}{ll} (\overline{N}K_{1})(wz), &{} \hbox {if }wz\in (\overline{N}K_{1})^* \hbox { but }wz\not \in (\overline{N}K_{2})^*; \\ (\overline{N}K_{2})(wz), &{} \hbox {if }wz\in (\overline{N}K_{2})^* \hbox { but }wz\not \in (\overline{N}K_{1})^*; \\ \max \{(\overline{N}K_{1})(wz), (\overline{N}K_{2})(wz)\}, &{} \hbox {if } wz\in (\overline{N}K_{1})^*\cap (\overline{N}K_{2})^*. \end{array} \right. \end{aligned}$$

Example 2.4

Let \(J^*=\{t_1,t_2,t_3\}\) be a set. Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*\), where \(\underline{\hat{G}}_{1}=(\underline{M}J_{1}, \underline{N}K_{1})\) and \(\overline{\hat{G}}_{1}=(\overline{M}J_{1}, \overline{N}K_{1})\) are FDs as shown in Fig. 3.
Fig. 3

Rough fuzzy digraph \(\hat{G}_1 =(\underline{\hat{G}}_1, \overline{\hat{G}}_1)\)

\(\underline{\hat{G}}_{2}=(\underline{M}J_{2}, \underline{N}K_{2})\) and \(\overline{\hat{G}}_{2}=(\overline{M}J_{2}, \overline{N}K_{2})\) are also FDs as shown in Fig. 4.
Fig. 4

Rough fuzzy digraph \(\hat{G}_2 =(\underline{\hat{G}}_2, \overline{\hat{G}}_2)\)

The union of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a \(\hat{G}=\hat{G}_{1}\Cup \hat{G}_{2}=(\underline{\hat{G}}_{1}\cup \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\cup \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\cup \underline{\hat{G}}_{2}=(\underline{M}J_{1}\cup \underline{M}J_{2}, \underline{N}K_{1}\cup \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\cup \overline{\hat{G}}_{2}=(\overline{M}J_{1}\cup \overline{M}J_{2}, \overline{N}K_{1}\cup \overline{N}K_{2})\) are FDs as shown in Fig. 5.

Fig. 5

Rough fuzzy digraph \(\hat{G}_1\Cup \hat{G}_2 =(\underline{\hat{G}}_1\cup \underline{\hat{G}}_2, \overline{\hat{G}}_1\cup \overline{\hat{G}}_2)\)

Theorem 2.1

The union of two RFDs is also a RFD.

Proof

Let \(\hat{G}_{1} = (\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_{2} = (\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Let \(\hat{G} = \hat{G}_{1} \Cup \hat{G}_{2} = (\underline{\hat{G}}_1\cup \underline{\hat{G}}_2, \overline{\hat{G}}_1\cup \overline{\hat{G}}_2)\) be the union of \(\hat{G}_{1}\) and \(\hat{G}_{2}\), where \(\underline{\hat{G}}_1\cup \underline{\hat{G}}_2 = (\underline{M}J_{1}\cup \underline{M}J_{2}, \underline{N}K_1\cup \underline{N}K_2)\) and \(\overline{\hat{G}}_1\cup \overline{\hat{G}}_2 = (\overline{M}J_{1}\cup \overline{M}J_{2}, \overline{N}K_1\cup \overline{N}K_2).\) We claim that \(\hat{G} = \hat{G}_{1} \Cup \hat{G}_{2}\) is a RFD. It is enough to show that \(\underline{N}K_1\cup \underline{N}K_2\) and \(\overline{N}K_1\cup \overline{N}K_2\) are fuzzy relations on \(\underline{M}J_{1}\cup \underline{M}J_{2}\) and \(\overline{M}J_{1}\cup \overline{M}J_{2},\) respectively. First, we show that \(\underline{N}K_1\cup \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\cup \underline{M}J_{2}\).

First we consider the case when \(wz\in (\underline{N}K_{1})^*\cap (\underline{N}K_2)^*\), then
$$\begin{aligned} (\underline{N}K_1\cup \underline{N}K_2)(wz)= & {} (\underline{N}K_1)(wz)\vee (\underline{N}K_2)(wz) \\\le & {} ((\underline{M}J_1)(w)\wedge (\underline{M}J_1)(z))\vee ((\underline{M}J_2)(w)\wedge (\underline{M}J_2)(z)) \\= & {} ((\underline{M}J_1)(w)\vee (\underline{M}J_2)(w))\wedge ((\underline{M}J_1)(z)\vee (\underline{M}J_2)(z)) \\= & {} (\underline{M}J_1\cup \underline{M}J_2)(w)\wedge (\underline{M}J_1\cup \underline{M}J_2)(z) \\ (\underline{N}K_1\cup \underline{N}K_2)(wz)\le & {} (\underline{M}J_1\cup \underline{M}J_2)(w)\wedge (\underline{M}J_1\cup \underline{M}J_2)(z) \end{aligned}$$
If \(wz\in (\underline{N}K_{1})^*\) and \(wz\not \in (\underline{N}K_{2})^*\), then
$$\begin{aligned} (\underline{N}K_1\cup \underline{N}K_2)(wz)= & {} (\underline{N}K_1)(wz) \\\le & {} (\underline{M}J_1)(w)\wedge (\underline{M}J_1)(z) \\= & {} (\underline{M}J_1\cup \underline{M}J_2)(w)\wedge (\underline{M}J_1\cup \underline{M}J_2)(z) \\ (\underline{N}K_1\cup \underline{N}K_2)(wz)\le & {} (\underline{M}J_1\cup \underline{M}J_2)(w)\wedge (\underline{M}J_1\cup \underline{M}J_2)(z) \end{aligned}$$
If \(wz\not \in (\underline{N}K_{1})^*\), but \(wz\in (\underline{N}K_{2})^*\), then
$$\begin{aligned} (\underline{N}K_1\cup \underline{N}K_2)(wz)= & {} (\underline{N}K_2)(wz) \\\le & {} (\underline{M}J_2)(w)\wedge (\underline{M}J_2)(z) \\= & {} (\underline{M}J_1\cup \underline{M}J_2)(w)\wedge (\underline{M}J_1\cup \underline{M}J_2)(z) \\ (\underline{N}K_1\cup \underline{N}K_2)(wz)\le & {} (\underline{M}J_1\cup \underline{M}J_2)(w)\wedge (\underline{M}J_1\cup \underline{M}J_2)(z) \end{aligned}$$
Thus, \(\underline{N}K_1\cup \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\cup \underline{M}J_{2}\).

Similarly, we can show that \(\overline{N}K_1\cup \overline{N}K_2\) is a fuzzy relation on \(\overline{M}J_{1}\cup \overline{M}J_{2}\). Hence, \(\hat{G}\) is a RFD. \(\square \)

Definition 2.7

Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*\). The intersection of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a RFD \(\hat{G}=\hat{G}_{1}\Cap \hat{G}_{2}=(\underline{\hat{G}}_{1}\cap \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\cap \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\cap \underline{\hat{G}}_{2}=(\underline{M}J_{1}\cap \underline{M}J_{2}, \underline{N}K_{1}\cap \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\cap \overline{\hat{G}}_{2}=(\overline{M}J_{1}\cap \overline{M}J_{2}, \overline{N}K_{1}\cap \overline{N}K_{2})\) are FDs, respectively, such that
$$\begin{aligned}&(\underline{M}J_{1}\cap \underline{M}J_{2})(w)\\&\quad = \left\{ \begin{array}{ll} (\underline{M}J_{1})(w), &{} \hbox {if }w\in (\underline{M}J_{1})^*\hbox { but }w\not \in (\underline{M}J_{2})^*; \\ (\underline{M}J_{2})(w), &{} \hbox {if }w\in (\underline{M}J_{2})^*\hbox { but }w\not \in (\underline{M}J_{1})^*; \\ \min \{(\underline{M}J_{1})(w), (\underline{M}J_{2})(w)\}, &{} \hbox {if } w\in (\underline{M}J_{1})^*\cap (\underline{M}J_{2})^*. \end{array} \right. \\&(\underline{N}K_{1}\cap \underline{N}K_{2})(wz)\\&\quad = \left\{ \begin{array}{ll} (\underline{N}K_{1})(wz), &{} \hbox {if }wz\in (\underline{N}K_{1})^* \hbox { but }wz\not \in (\underline{N}K_{2})^*; \\ (\underline{N}K_{2})(wz), &{} \hbox {if }wz\in (\underline{N}K_{2})^* \hbox { but }wz\not \in (\underline{N}K_{1})^*; \\ \min \{(\underline{N}K_{1})(wz), (\underline{N}K_{2})(wz)\}, &{} \hbox {if }wz\in (\underline{N}K_{1})^*\cap (\underline{N}K_{2})^*. \end{array} \right. \\&(\overline{M}J_{1}\cap \overline{M}J_{2})(w)\\&\quad = \left\{ \begin{array}{ll} (\overline{M}J_{1})(w), &{} \hbox {if }w\in (\overline{M}J_{1})^*\hbox { but } w\not \in (\overline{M}J_{2})^*; \\ (\overline{M}J_{2})(w), &{} \hbox {if }w\in (\overline{M}J_{2})^*\hbox { but } w\not \in (\overline{M}J_{1})^*; \\ \min \{(\overline{M}J_{1})(w), (\overline{M}J_{2})(w)\}, &{} \hbox {if } w\in (\overline{M}J_{1})^*\cap (\overline{M}J_{2})^*. \end{array} \right. \\&(\overline{N}K_{1}\cap \overline{N}K_{2})(wz)\\&\quad = \left\{ \begin{array}{ll} (\overline{N}K_{1})(wz), &{} \hbox {if }wz\in (\overline{N}K_{1})^* \hbox { but }wz\not \in (\overline{N}K_{2})^*; \\ (\overline{N}K_{2})(wz), &{} \hbox {if }wz\in (\overline{N}K_{2})^* \hbox { but }wz\not \in (\overline{N}K_{1})^*; \\ \min \{(\overline{N}K_{1})(wz), (\overline{N}K_{2})(wz)\}, &{} \hbox {if } wz\in (\overline{N}K_{1})^*\cap (\overline{N}K_{2})^*. \end{array} \right. \end{aligned}$$

Example 2.5

Consider the two RFDs \(\hat{G}_{1}\) and \(\hat{G}_2\) as shown in Figs. 3 and 4.

The intersection of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is \(\hat{G}=\hat{G}_{1}\Cap \hat{G}_{2}=(\underline{\hat{G}}_{1}\cap \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\cap \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\cap \underline{\hat{G}}_{2}=(\underline{M}J_{1}\cap \underline{M}J_{2}, \underline{N}K_{1}\cap \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\cap \overline{\hat{G}}_{2}=(\overline{M}J_{1}\cap \overline{M}J_{2}, \overline{N}K_{1}\cap \overline{N}K_{2})\) are FDs as shown in Fig. 6.

Fig. 6

Rough fuzzy digraph \(\hat{G}_1\Cap \hat{G}_2 =(\underline{\hat{G}}_1\cap \underline{\hat{G}}_2, \overline{\hat{G}}_1\cap \overline{\hat{G}}_2)\)

Definition 2.8

Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two direct sum of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is \(\hat{G}_{1}\widehat{\oplus } \hat{G}_{2}=(\underline{\hat{G}}_{1}\widehat{\oplus } \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\widehat{\oplus } \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\widehat{\oplus } \underline{\hat{G}}_{2}=(\underline{M}J_{1}\widehat{\oplus } \underline{M}J_{2}, \underline{N}K_{1}\widehat{\oplus } \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\widehat{\oplus } \overline{\hat{G}}_{2}=(\overline{M}J_{1}\widehat{\oplus } \overline{M}J_{2}, \overline{N}K_{1}\widehat{\oplus } \overline{N}K_{2})\) are fuzzy digraphs, respectively, such that
$$\begin{aligned}&(\underline{M}J_{1}\widehat{\oplus } \underline{M}J_{2})(w)\\&\quad = \left\{ \begin{array}{ll} (\underline{M}J_{1})(w), &{} \hbox {if }w\in (\underline{M}J_{1})^* \hbox { but }w\not \in (\underline{M}J_{2})^*; \\ (\underline{M}J_{2})(w), &{} \hbox {if }w\in (\underline{M}J_{2})^* \hbox { but }w\not \in (\underline{M}J_{1})^*; \\ \max \{(\underline{M}J_{1})(w), (\underline{M}J_{2})(w)\}, &{} \hbox {if } w\in (\underline{M}J_{1})^*\cap (\underline{M}J_{2})^*. \end{array} \right. \\&(\underline{N}K_{1}\widehat{\oplus } \underline{N}K_{2})(wz)\\&\quad = \left\{ \begin{array}{ll} (\underline{N}K_{1})(wz), &{} \hbox {if }wz\in (\underline{N}K_{1})^* \hbox { but }wz\not \in (\underline{N}K_{2})^*; \\ (\underline{N}K_{2})(wz), &{} \hbox {if }wz\in \underline{N}K_{2}\hbox { but } wz\not \in (\underline{N}K_{1})^*; \\ ~~~~~~0~~~~~~~, &{} \hbox {if }wz\in (\underline{N}K_{1})^*\cap (\underline{N}K_{2})^*. \end{array} \right. \\&(\overline{M}J_{1}\widehat{\oplus } \overline{M}J_{2})(w)\\&\quad = \left\{ \begin{array}{ll} (\overline{M}J_{1})(w), &{} \hbox {if }w\in (\overline{M}J_{1})^*\hbox { but } w\not \in (\overline{M}J_{2})^*; \\ (\overline{M}J_{2})(w), &{} \hbox {if } w\in (\overline{M}J_{2})^*\hbox { but }w\not \in (\overline{M}J_{1})^*; \\ \max \{(\overline{M}J_{1})(w), (\overline{M}J_{2})(w)\}, &{} \hbox {if } w\in (\overline{M}J_{1})^*\cap (\overline{M}J_{2})^*. \end{array} \right. \\&(\overline{N}K_{1}\widehat{\oplus } \overline{N}K_{2})(wz)\\&\quad = \left\{ \begin{array}{ll} (\overline{N}K_{1})(wz), &{} \hbox {if }wz\in (\overline{N}K_{1})^* \hbox { but }wz\not \in (\overline{N}K_{2})^*; \\ (\overline{N}K_{2})(wz), &{} \hbox {if }wz\in (\overline{N}K_{2})^* \hbox { but }wz\not \in (\overline{N}K_{1})^*; \\ ~~~~~~0~~~~~~~, &{} \hbox {if }wz\in (\overline{N}K_{1})^*\cap (\overline{N}K_{2})^*. \end{array} \right. \end{aligned}$$

Example 2.6

Let \(J^*=\{t_1,t_2,t_3,t_4\}\) be a set of universe. Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*\), where \(\underline{\hat{G}}_{1}=(\underline{M}J_{1}, \underline{N}K_{1})\) and \(\overline{\hat{G}}_{1}=(\overline{M}J_{1}, \overline{N}K_{1})\) are fuzzy digraphs as shown in Fig. 7.
Fig. 7

Rough fuzzy digraph \(\hat{G}_1 =(\underline{\hat{G}}_1, \overline{\hat{G}}_1)\)

\(\underline{\hat{G}}_{2}=(\underline{M}J_{2}, \underline{N}K_{2})\) and \(\overline{\hat{G}}_{2}=(\overline{M}J_{2}, \overline{N}K_{2})\) are also fuzzy digraphs as shown in Fig. 8.
Fig. 8

Rough fuzzy digraph \(\hat{G}_2 =(\underline{\hat{G}}_2, \overline{\hat{G}}_2)\)

The direct sum of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is \(\hat{G}_{1}\widehat{\oplus } \hat{G}_{2}=(\underline{\hat{G}}_{1}\widehat{\oplus }\underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\widehat{\oplus }\overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\widehat{\oplus }\underline{\hat{G}}_{2}=(\underline{M}J_{1}\widehat{\oplus }\underline{M}J_{2}, \underline{N}K_{1}\widehat{\oplus }\underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\widehat{\oplus }\overline{\hat{G}}_{2}=(\overline{M}J_{1}\widehat{\oplus }\overline{M}J_{2}, \overline{N}K_{1}\widehat{\oplus }\overline{N}K_{2})\) are fuzzy digraphs as shown in Fig. 9.

Hence, \(\hat{G}_{1}\widehat{\oplus }\hat{G}_{2}\) is a RFD.

Fig. 9

\(\hat{G}_1\widehat{\oplus } \hat{G}_2 =(\underline{\hat{G}}_1\widehat{\oplus }\underline{\hat{G}}_2, \overline{\hat{G}}_1\widehat{\oplus } \overline{\hat{G}}_2)\)

Remark

The direct sum of two RFDs need not to be a RFD.

Definition 2.9

The Cartesian product of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a RFD \(\hat{G} = \hat{G}_{1} \ltimes \hat{G}_{2} = (\underline{\hat{G}}_{1} \ltimes \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1} \ltimes \overline{\hat{G}}_{2}),\) where \(\underline{\hat{G}}_{1}\ltimes \underline{\hat{G}}_{2}=(\underline{M}J_{1}\ltimes \underline{M}J_{2}, \underline{N}K_{1}\ltimes \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\ltimes \overline{\hat{G}}_{2}=(\overline{M}J_{1}\ltimes \overline{M}J_{2}, \overline{N}K_{1}\ltimes \overline{N}K_{2})\) are FDs, respectively, such that
  1. (i)

    \((\underline{M}J_{1}\ltimes \underline{M}J_{2})(w_1,w_2) = \min \{(\underline{M}J_{1})(w_1), (\underline{M}J_{2})(w_2)\},~~ \forall ~(w_1,w_2)\in (\underline{M}J_{1})^*\times (\underline{M}J_{2})^*,\)

    \((\underline{N}K_{1}\ltimes \underline{N}K_{2})\big ((w,w_2)(w,z_2)\big ) = \min \{(\underline{M}J_{1})(w), (\underline{N}K_{2})(w_2z_2)\},~~ \forall ~w\in (\underline{M}J_1)^*, w_2z_2\in (\underline{N}K_2)^*\).

    \((\underline{N}K_{1}\ltimes \underline{N}K_{2})\big ((w_1,z)(z_1,z)\big ) = \min \{(\underline{N}K_{1})(w_1z_1), (\underline{M}J_{2})(z)\},~~ \forall ~w_1z_1\in (\underline{N}K_1)^*, z\in (\underline{M}J_2)^*\).

     
  2. (ii)

    \((\overline{M}J_{1}\ltimes \overline{M}J_{2})(w_1,w_2) = \min \{(\overline{M}J_{1})(w_1), (\overline{M}J_{2})(w_2)\},~~ \forall ~(w_1,w_2)\in (\overline{M}J_{1})^*\times (\overline{M}J_{2})^*,\)

    \((\overline{N}K_{1}\ltimes \overline{N}K_{2})\big ((w,w_2)(w,z_2)\big ) = \min \{(\overline{M}J_{1})(w), (\overline{N}K_{2})(w_2z_2)\},~~ \forall ~w\in (\overline{M}J_{1})^*, w_2z_2\in (\overline{N}K_{2})^*\).

    \((\overline{N}K_{1}\ltimes \overline{N}K_{2})\big ((w_1,z)(z_1,z)\big ) = \min \{(\overline{N}K_{1})(w_1z_1), (\overline{M}J_{2})(z)\},~~ \forall ~w_1z_1\in (\overline{N}K_{1})^*, z\in (\overline{M}J_{2})^*\).

     

Example 2.7

Consider the two RFDs \(\hat{G}_{1}\) and \(\hat{G}_2\) as shown in Figs. 3 and 4.

The Cartesian product of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is \(\hat{G}=\hat{G}_{1}\ltimes \hat{G}_{2}=(\underline{\hat{G}}_{1}\ltimes \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\ltimes \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\ltimes \underline{\hat{G}}_{2}=(\underline{M}J_{1}\ltimes \underline{M}J_{2}, \underline{N}K_{1}\ltimes \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\ltimes \overline{\hat{G}}_{2}=(\overline{M}J_{1}\ltimes \overline{M}J_{2}, \overline{N}K_{1}\ltimes \overline{N}K_{2})\) are FDs as shown in Fig. 10.

Fig. 10

Rough fuzzy digraph \(\hat{G}_1\ltimes \hat{G}_2 =(\underline{\hat{G}}_1\ltimes \underline{\hat{G}}_2, \overline{\hat{G}}_1\ltimes \overline{\hat{G}}_2)\)

Theorem 2.2

The Cartesian product of two RFDs is also a RFD.

Proof

Let \(\hat{G}_{1} = (\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_{2} = (\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Let \(\hat{G} = \hat{G}_{1} \ltimes \hat{G}_{2} = (\underline{\hat{G}}_1\ltimes \underline{\hat{G}}_2, \overline{\hat{G}}_1\ltimes \overline{\hat{G}}_2)\) be the Cartesian product of \(\hat{G}_{1}\) and \(\hat{G}_{2}\), where \(\underline{\hat{G}}_1\ltimes \underline{\hat{G}}_2 = (\underline{M}J_{1}\ltimes \underline{M}J_{2}, \underline{N}K_1\ltimes \underline{N}K_2)\) and \(\overline{\hat{G}}_1\ltimes \overline{\hat{G}}_2 = (\overline{M}J_{1}\ltimes \overline{M}J_{2}, \overline{N}K_1\ltimes \overline{N}K_2).\) We claim that \(\hat{G} = \hat{G}_{1} \ltimes \hat{G}_{2}\) is a RFD. It is enough to show that \(\underline{N}K_1\ltimes \underline{N}K_2\) and \(\overline{N}K_1\ltimes \overline{N}K_2\) are fuzzy relations on \(\underline{M}J_{1}\ltimes \underline{M}J_{2}\) and \(\overline{M}J_{1}\ltimes \overline{M}J_{2},\) respectively. First, we show that \(\underline{N}K_1\ltimes \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\ltimes \underline{M}J_{2}\).

If \(w\in (\underline{M}J_{1})^*\), \(w_2z_2\in (\underline{N}K_{2})^*\), then
$$\begin{aligned} (\underline{N}K_1\ltimes \underline{N}K_2)((w,w_2)(w,z_2))= & {} (\underline{M}J_1)(w)\wedge (\underline{N}K_2)(w_2z_2) \\\le & {} (\underline{M}J_1)(w)\wedge ((\underline{M}J_2)(w_2)\wedge (\underline{M}J_2)(z_2)) \\= & {} ((\underline{M}J_1)(w)\wedge (\underline{M}J_2)(w_2))\wedge ((\underline{M}J_1)(w)\wedge (\underline{M}J_2)(z_2)) \\= & {} (\underline{M}J_1\ltimes \underline{M}J_2)(w,w_2)\wedge (\underline{M}J_1\ltimes \underline{M}J_2)(w,z_2) \\ (\underline{N}K_1\ltimes \underline{N}K_2)((w,w_2)(w,z_2))\le & {} (\underline{M}J_1\ltimes \underline{M}J_2)(w,w_2)\wedge (\underline{M}J_1\ltimes \underline{M}J_2)(w,z_2) \end{aligned}$$
If \(w_1z_1\in (\underline{N}K_{1})^*\), \(z\in (\underline{M}J_{2})^*\), then
$$\begin{aligned} (\underline{N}K_1\ltimes \underline{N}K_2)((w_1,z)(z_1,z))= & {} (\underline{N}K_1)(w_1z_1)\wedge (\underline{M}J_2)(z) \\\le & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_1)(z_1))\wedge (\underline{M}J_2)(z) \\= & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_2)(z))\wedge ((\underline{M}J_1)(z_1)\wedge (\underline{M}J_2)(z)) \\= & {} (\underline{M}J_1\ltimes \underline{M}J_2)(w_1,z)\wedge (\underline{M}J_1\ltimes \underline{M}J_2)(z_1,z) \\ (\underline{N}K_1\ltimes \underline{N}K_2)((w_1,z)(z_1,z))\le & {} (\underline{M}J_1\ltimes \underline{M}J_2)(w_1,z)\wedge (\underline{M}J_1\ltimes \underline{M}J_2)(z_1,z) \end{aligned}$$
Thus, \(\underline{N}K_1\ltimes \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\ltimes \underline{M}J_{2}\).

Similarly, \(\overline{N}K_1\ltimes \overline{N}K_2\) is a fuzzy relation on \(\overline{M}J_{1}\ltimes \overline{M}J_{2}\). Hence, \(\hat{G}\) is a RFD. \(\square \)

Definition 2.10

The maximal product of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a RFD \(\hat{G} = \hat{G}_{1}*\hat{G}_{2} = (\underline{\hat{G}}_{1} *\underline{\hat{G}}_{2}, \overline{\hat{G}}_{1} *\overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}*\underline{\hat{G}}_{2}=(\underline{M}J_{1}*\underline{M}J_{2}, \underline{N}K_{1}*\underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}*\overline{\hat{G}}_{2}=(\overline{M}J_{1}*\overline{M}J_{2}, \overline{N}K_{1}*\overline{N}K_{2})\) are fuzzy digraphs, respectively, such that
  1. (i)

    \((\underline{M}J_{1}*\underline{M}J_{2})(w_1, w_{2}) = \max \{(\underline{M}J_{1})(w_1), (\underline{M}J_{2})(w_2)\},~~ \forall ~(w_1, w_{2})\in (\underline{M}J_{1})^*\times (\underline{M}J_{2})^*,\)

    \((\underline{N}K_{1}*\underline{N}K_{2})\big ((w, w_{2})(y,z_2)\big ) = \max \{(\underline{M}J_{1})(w), (\underline{N}K_{2})(w_2z_2)\},~~ \forall ~w\in (\underline{M}J_1)^*, w_2z_2\in (\underline{N}K_2)^*\),

    \((\underline{N}K_{1}*\underline{N}K_{2})\big ((w_1, z)(z_1,z)\big ) = \max \{(\underline{N}K_{1})(w_1z_1), (\underline{M}J_{2})(z)\},~~ \forall ~w_1z_1\in (\underline{N}K_1)^*, z\in (\underline{M}J_2)^*\).

     
  2. (ii)

    \((\overline{M}J_{1}*\overline{M}J_{2})(w_1, w_{2}) = \max \{(\overline{M}J_{1})(w_1), (\overline{M}J_{2})(w_2)\},~~ \forall ~(w_1, w_{2})\in (\overline{M}J_{1})^*\times (\overline{M}J_{2})^*,\)

    \((\overline{N}K_{1}*\overline{N}K_{2})\big ((w, w_{2})(y,z_2)\big ) = \max \{(\overline{M}J_{1})(w), (\overline{N}K_{2})(w_2z_2)\},~~ \forall ~w\in (\overline{M}J_1)^*, w_2z_2\in (\overline{N}K_2)^*\),

    \((\overline{N}K_{1}*\overline{N}K_{2})\big ((w_1, z)(z_1,z)\big ) = \max \{(\overline{N}K_{1})(w_1z_1), (\overline{M}J_{2})(z)\},~~ \forall ~w_1z_1\in (\overline{N}K_1)^*, z\in (\overline{M}J_2)^*\).

     

Example 2.8

Consider the two RFDs \(\hat{G}_{1}\) and \(\hat{G}_2\) as shown in Figs. 3 and 4.

The maximal product of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is \(\hat{G}=\hat{G}_{1}*\hat{G}_{2}=(\underline{\hat{G}}_{1}*\underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}*\overline{\hat{G}}_{2})\), as shown in Fig. 11.

Thus, \(\hat{G}\) is a RFD.

Fig. 11

Rough fuzzy digraph \(\hat{G}_1*\hat{G}_2 =(\underline{\hat{G}}_1*\underline{\hat{G}}_2, \overline{\hat{G}}_1*\overline{\hat{G}}_2)\)

Proposition 2.1

Let \(\hat{G}_{1}\) and \(\hat{G}_{2}\) be two RFDs. Then their maximal product \(\hat{G}_{1}*\hat{G}_{2}\) is a RFD.

Proof

Let \(\hat{G} = \hat{G}_{1} *\hat{G}_{2} = (\underline{\hat{G}}_1*\underline{\hat{G}}_2, \overline{\hat{G}}_1*\overline{\hat{G}}_2)\), where \(\underline{\hat{G}}_1*\underline{\hat{G}}_2 = (\underline{M}J_{1}*\underline{M}J_{2}, \underline{N}K_1*\underline{N}K_2)\) and \(\overline{\hat{G}}_1*\overline{\hat{G}}_2 = (\overline{M}J_{1}*\overline{M}J_{2}, \overline{N}K_1*\overline{N}K_2).\) We claim that \(\hat{G} = \hat{G}_{1} *\hat{G}_{2}\) is a RFD. It is enough to show that \(\underline{N}K_1*\underline{N}K_2\) and \(\overline{N}K_1*\overline{N}K_2\) are fuzzy relations on \(\underline{M}J_{1}*\underline{M}J_{2}\) and \(\overline{M}J_{1}*\overline{M}J_{2},\) respectively. First, we show that \(\underline{N}K_1*\underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}*\underline{M}J_{2}\).

If \(w\in (\underline{M}J_1)^*\), \(w_2z_2\in (\underline{N}K_2)^*\), then
$$\begin{aligned} (\underline{N}K_1*\underline{N}K_2)((w,w_2)(w,z_2))= & {} (\underline{M}J_1)(w)\vee (\underline{N}K_2)(w_2z_2) \\\le & {} (\underline{M}J_1)(w)\vee ((\underline{M}J_2)(w_2)\wedge (\underline{M}J_2)(z_2)) \\= & {} ((\underline{M}J_1)(w)\vee (\underline{M}J_2)(w_2))\wedge ((\underline{M}J_1)(w)\vee (\underline{M}J_2)(z_2)) \\= & {} (\underline{M}J_1*\underline{M}J_2)(w,w_2)\wedge (\underline{M}J_1*\underline{M}J_2)(w,z_2) \\ (\underline{N}K_1*\underline{N}K_2)((w,w_2)(w,z_2))\le & {} (\underline{M}J_1*\underline{M}J_2)(w,w_2)\wedge (\underline{M}J_1*\underline{M}J_2)(w,z_2) \end{aligned}$$
If \(w_1z_1\in (\underline{N}K_1)^*\), \(z\in (\underline{M}J_2)^*\), then
$$\begin{aligned} (\underline{N}K_1*\underline{N}K_2)((w_1,z)(z_1,z))= & {} (\underline{N}K_1)(w_1z_1)\vee (\underline{M}J_2)(z) \\\le & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_1)(z_1))\vee (\underline{M}J_2)(z) \\= & {} ((\underline{M}J_1)(w_1)\vee (\underline{M}J_2)(z))\wedge ((\underline{M}J_1)(z_1)\vee (\underline{M}J_2)(z)) \\= & {} (\underline{M}J_1*\underline{M}J_2)(w_1,z)\wedge (\underline{M}J_1*\underline{M}J_2)(z_1,z) \\ (\underline{N}K_1*\underline{N}K_2)((w_1,z)(z_1,z))\le & {} (\underline{M}J_1*\underline{M}J_2)(w_1,z)\wedge (\underline{M}J_1*\underline{M}J_2)(z_1,z) \end{aligned}$$
Thus, \(\underline{N}K_1*\underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}*\underline{M}J_{2}\).

Similarly, \(\overline{N}K_1*\overline{N}K_2\) is a fuzzy relation on \(\overline{M}J_{1}*\overline{M}J_{2}\). Hence, \(\hat{G}\) is a RFD. \(\square \)

Definition 2.11

The residue product of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a RFD \(\hat{G} = \hat{G}_{1}\bullet \hat{G}_{2} = (\underline{\hat{G}}_{1}\bullet \underline{\hat{G}}_{2}, \overline{\hat{G}}_{2}),\) where \(\underline{\hat{G}}_{1}\bullet \underline{\hat{G}}_{2}=(\underline{M}J_{1}\bullet \underline{M}J_{2}, \underline{N}K_{1}\bullet \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\bullet \overline{\hat{G}}_{2}=(\overline{M}J_{1}\bullet \overline{M}J_{2}, \overline{N}K_{1}\bullet \overline{N}K_{2})\) are fuzzy digraphs, respectively, such that
  1. (i)

    \((\underline{M}J_{1}\bullet \underline{M}J_{2})(w_1, w_{2}) = \max \{(\underline{M}J_{1})(w_1), (\underline{M}J_{2})(w_2)\},~~ \forall ~(w_1, w_{2})\in (\underline{M}J_{1})^*\times (\underline{M}J_{2})^*,\)

    \((\underline{N}K_{1}\bullet \underline{N}K_{2})\big ((w_1, w_{2})(z_1,z_2)\big ) = (\underline{N}K_{1})(w_1z_1),~~ \forall ~w_1z_1\in (\underline{N}K_1)^*,~w_{2}, z_{2}\in (\underline{M}J_2)^*\) such that \(w_{2}\ne z_{2}\).

     
  2. (ii)

    \((\overline{M}J_{1}\bullet \overline{M}J_{2})(w_1, w_{2}) = \max \{(\overline{M}J_{1})(w_1), (\overline{M}J_{2})(w_2)\},~~ \forall ~(w_1, w_{2})\in (\overline{M}J_{1})^*\times (\overline{M}J_{2})^*,\)

    \((\overline{N}K_{1}\bullet \overline{N}K_{2})\big ((w_1, w_{2})(z_1,z_2)\big ) = (\overline{N}K_{1})(w_1z_1),~~ \forall ~w_1z_1\in (\overline{N}K_1)^*,~w_{2}, z_{2}\in (\overline{M}J_2)^*\) such that \(w_{2}\ne z_{2}\).

     

Proposition 2.2

Let \(\hat{G}_{1}\) and \(\hat{G}_{2}\) be two RFDs. Then their residue product \(\hat{G}_{1}\bullet \hat{G}_{2}\) is a RFD.

Proof

Let \(\hat{G} = \hat{G}_{1}\bullet \hat{G}_{2} = (\underline{\hat{G}}_1\bullet \underline{\hat{G}}_2, \overline{\hat{G}}_1\bullet \overline{\hat{G}}_2)\), where \(\underline{\hat{G}}_1\bullet \underline{\hat{G}}_2 = (\underline{M}J_{1}\bullet \underline{M}J_{2}, \underline{N}K_1\bullet \underline{N}K_2)\) and \(\overline{\hat{G}}_1\bullet \overline{\hat{G}}_2 = (\overline{M}J_{1}\bullet \overline{M}J_{2}, \overline{N}K_1\bullet \overline{N}K_2).\) We claim that \(\hat{G} = \hat{G}_{1}\bullet \hat{G}_{2}\) is a RFD. It is enough to show that \(\underline{N}K_1\bullet \underline{N}K_2\) and \(\overline{N}K_1\bullet \overline{N}K_2\) are fuzzy relations on \(\underline{M}J_{1}\bullet \underline{M}J_{2}\) and \(\overline{M}J_{1}\bullet \overline{M}J_{2},\) respectively. First, we show that \(\underline{N}K_1\bullet \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\bullet \underline{M}J_{2}\).

If \(w_1z_1\in (\underline{N}K_1)^*\), \(w_2, z_2\in (\underline{M}J_2)^*\) such that \(w_2\ne z_2\), then
$$\begin{aligned} (\underline{N}K_1\bullet \underline{N}K_2)((w_1,w_2)(z_1,z_2))= & {} (\underline{N}K_1)(w_1z_1)\\\le & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_1)(z_1)) \\= & {} ((\underline{M}J_1)(w_1)\vee (\underline{M}J_2)(w_2))\wedge ((\underline{M}J_1)(z_1)\vee (\underline{M}J_2)(z_2)) \\= & {} (\underline{M}J_1\bullet \underline{M}J_2)(w_1,w_2)\wedge (\underline{M}J_1\bullet \underline{M}J_2)(z_1,z_2) \\ (\underline{N}K_1\bullet \underline{N}K_2)((w_1,w_2)(z_1,z_2))\le & {} (\underline{M}J_1\bullet \underline{M}J_2)(w_1,w_2)\wedge (\underline{M}J_1\bullet \underline{M}J_2)(z_1,z_2) \end{aligned}$$
Thus, \(\underline{N}K_1\bullet \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\bullet \underline{M}J_{2}\).

Similarly, we can show that \(\overline{N}K_1\bullet \overline{N}K_2\) is a fuzzy relation on \(\overline{M}J_{1}\bullet \overline{M}J_{2}\). Hence, \(\hat{G}\) is a RFD. \(\square \)

Definition 2.12

The composition of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is a RFD \(\hat{G} = \hat{G}_{1}[\hat{G}_{2}] = (\underline{\hat{G}}_{1} \times \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1} \times \overline{\hat{G}}_{2}),\) where \(\underline{\hat{G}}_{1}\times \underline{\hat{G}}_{2}=(\underline{M}J_{1}\times \underline{M}J_{2}, \underline{N}K_{1}\times \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\times \overline{\hat{G}}_{2}=(\overline{M}J_{1}\times \overline{M}J_{2}, \overline{N}K_{1}\times \overline{N}K_{2})\) are FDs, respectively, such that
  1. (i)

    \((\underline{M}J_{1}\times \underline{M}J_{2})(w_1,w_2) = \min \{(\underline{M}J_{1})(w_1), (\underline{M}J_{2})(w_2)\},~~ \forall ~(w_1,w_2)\in (\underline{M}J_{1})^*\times (\underline{M}J_{2})^*,\)

    \((\underline{N}K_{1}\times \underline{N}K_{2})\big ((w,w_2)(w,z_2)\big ) = \min \{(\underline{M}J_{1})(w), (\underline{N}K_{2})(w_2z_2)\},~~ \forall ~w\in (\underline{M}J_{1})^*, w_2z_2\in (\underline{N}K_{2})^*,\)

    \((\underline{N}K_{1}\times \underline{N}K_{2})\big ((w_1,z)(z_1,z)\big ) = \min \{(\underline{N}K_{1})(w_1z_1), (\underline{M}J_{2})(z)\},~~ \forall ~w_1z_1\in (\underline{N}K_{1})^*, z\in (\underline{M}J_{2})^*,\)

    \((\underline{N}K_{1}\times \underline{N}K_{2})\big ((w_1,w_2)(z_1,z_2)\big ) = \min \{\underline{N}K_{1})(w_1z_1), (\underline{M}J_{2})(w_2), (\underline{M}J_{2})(z_2)\}, \forall ~w_1z_1\in (\underline{N}K_1)^*,~w_2, z_2\in (\underline{M}J_{2})^*\) such that \(w_2\ne z_2\).

     
  2. (ii)

    \((\overline{M}J_{1}\times \overline{M}J_{2})(w_1,w_2) = \min \{(\overline{M}J_{1})(w_1), (\overline{M}J_{2})(w_2)\},~~ \forall ~(w_1,w_2)\in (\overline{M}J_{1})^*\times (\overline{M}J_{2})^*,\)

    \((\overline{N}K_{1}\times \overline{N}K_{2})\big ((w,w_2)(w,z_2)\big ) = \min \{(\overline{M}J_{1})(w), (\overline{N}K_{2})(w_2z_2)\},~~ \forall ~w\in (\overline{M}J_{1})^*, w_2z_2\in (\overline{N}K_{2})^*,\)

    \((\overline{N}K_{1}\times \overline{N}K_{2})\big ((w_1,z)(z_1,z)\big ) = \min \{(\overline{N}K_{1})(w_1z_1), (\overline{M}J_{2})(z)\},~~ \forall ~w_1z_1\in (\overline{N}K_{1})^*, z\in (\overline{M}J_{2})^*,\)

    \((\overline{N}K_{1}\times \overline{N}K_{2})\big ((w_1,w_2)(z_1,z_2)\big ) = \min \{\overline{N}K_{1})(w_1z_1), (\overline{M}J_{2})(w_2), (\overline{M}J_{2})(z_2)\}, \forall ~w_1z_1\in (\overline{N}K_1)^*,~w_2, z_2\in (\overline{M}J_{2})^*\) such that \(w_2\ne z_2\).

     

Example 2.9

Consider the two RFDs \(\hat{G}_{1}\) and \(\hat{G}_2\) as shown in Figs. 3 and 4. The composition of \(\hat{G}_{1}\) and \(\hat{G}_{2}\) is \(\hat{G}=\hat{G}_{1}\times \hat{G}_{2}=(\underline{\hat{G}}_{1}\times \underline{\hat{G}}_{2}, \overline{\hat{G}}_{1}\times \overline{\hat{G}}_{2})\), where \(\underline{\hat{G}}_{1}\times \underline{\hat{G}}_{2}=(\underline{M}J_{1}\times \underline{M}J_{2}, \underline{N}K_{1}\times \underline{N}K_{2})\) and \(\overline{\hat{G}}_{1}\times \overline{\hat{G}}_{2}=(\overline{M}J_{1}\times \overline{M}J_{2}, \overline{N}K_{1}\times \overline{N}K_{2})\) are FDs as shown in Fig. 12.

Fig. 12

Rough fuzzy digraph \(\hat{G}_1\times \hat{G}_2 =(\underline{\hat{G}}_1\times \underline{\hat{G}}_2, \overline{\hat{G}}_1\times \overline{\hat{G}}_2)\)

Theorem 2.3

The composition of two RFDs is also a RFD.

Proof

Let \(\hat{G}_{1} = (\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_{2} = (\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Let \(\hat{G} = \hat{G}_{1} \times G_{2} = (\underline{\hat{G}}_1\times \underline{\hat{G}}_2, \overline{\hat{G}}_1\times \overline{\hat{G}}_2)\) be the composition of \(\hat{G}_{1} \) and \(\hat{G}_{2}\), where \(\underline{\hat{G}}_1\times \underline{\hat{G}}_2 = (\underline{M}J_{1}\times \underline{M}J_{2}, \underline{N}K_1\times \underline{N}K_2)\) and \(\overline{\hat{G}}_1\times \overline{\hat{G}}_2 = (\overline{M}J_{1}\times \overline{M}J_{2}, \overline{N}K_1\times \overline{N}K_2).\) We claim that \(\hat{G} = \hat{G}_{1} \times \hat{G}_{2}\) is a RFD. It is enough to show that \(\underline{N}K_1\times \underline{N}K_2\) and \(\overline{N}K_1\times \overline{N}K_2\) are fuzzy relations on \(\underline{M}J_{1}\times \underline{M}J_{2}\) and \(\overline{M}J_{1}\times \overline{M}J_{2},\) respectively. First, we show that \(\underline{N}K_1\times \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\times \underline{M}J_{2}\).

If \(x\in (\underline{M}J_{1})^*\), \(w_2z_2\in (\underline{N}K_{2})^*\), then
$$\begin{aligned} (\underline{N}K_1\times \underline{N}K_2)((w,w_2)(w,z_2))= & {} (\underline{M}J_1)(w)\wedge (\underline{N}K_2)(w_2z_2) \\\le & {} (\underline{M}J_1)(w)\wedge ((\underline{M}J_2)(w_2)\wedge (\underline{M}J_2)(z_2)) \\= & {} ((\underline{M}J_1)(w)\wedge (\underline{M}J_2)(w_2))\wedge ((\underline{M}J_1)(w)\wedge (\underline{M}J_2)(z_2)) \\= & {} (\underline{M}J_1\times \underline{M}J_2)(w,w_2)\wedge (\underline{M}J_1\times \underline{M}J_2)(w,z_2) \\ (\underline{N}K_1\times \underline{N}K_2)((w,w_2)(w,z_2))\le & {} (\underline{M}J_1\times \underline{M}J_2)(w,w_2)\wedge (\underline{M}J_1\times \underline{M}J_2)(w,z_2) \end{aligned}$$
If \(w_1z_1\in (\underline{N}K_{1})^*\), \(z\in (\underline{M}J_{2})^*\), then
$$\begin{aligned} (\underline{N}K_1\times \underline{N}K_2)((w_1,z)(z_1,z))= & {} (\underline{N}K_1)(w_1z_1)\wedge (\underline{M}J_2)(z) \\\le & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_1)(z_1))\wedge (\underline{M}J_2)(z) \\= & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_2)(z))\wedge ((\underline{M}J_1)(z_1)\wedge (\underline{M}J_2)(z)) \\= & {} (\underline{M}J_1\times \underline{M}J_2)(w_1,z)\wedge (\underline{M}J_1\times \underline{M}J_2)(z_1,z)\\ (\underline{N}K_1\times \underline{N}K_2)((w_1,z)(z_1,z))\le & {} (\underline{M}J_1\times \underline{M}J_2)(w_1,z)\wedge (\underline{M}J_1\times \underline{M}J_2)(z_1,z) \end{aligned}$$
If \(w_1z_1\in (\underline{N}K_{1})^*\), \(w_2, z_2\in (\underline{M}J_{2})^*\) such that \(w_2\ne z_2\), then
$$\begin{aligned} (\underline{N}K_1\times \underline{N}K_2)((w_1,w_2)(z_1,z_2))= & {} (\underline{N}K_1)(w_1z_1)\wedge (\underline{M}J_2)(w_2)\wedge (\underline{M}J_2)(z_2) \\\le & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_1)(z_1))\wedge (\underline{M}J_2)(w_2)\wedge (\underline{M}J_2)(z_2) \\= & {} ((\underline{M}J_1)(w_1)\wedge (\underline{M}J_1)(w_2))\wedge ((\underline{M}J_1)(z_1)\wedge (\underline{M}J_2)(z_2)) \\= & {} (\underline{M}J_1\times \underline{M}J_2)(w_1,w_2)\wedge (\underline{M}J_1\times \underline{M}J_2)(z_1,z_2) \\ (\underline{N}K_1\times \underline{N}K_2)((w_1,w_2)(z_1,z_2))\le & {} (\underline{M}J_1\times \underline{M}J_2)(w_1,w_2)\wedge (\underline{M}J_1\times \underline{M}J_2)(z_1,z_2) \end{aligned}$$
Thus, \(\underline{N}K_1\times \underline{N}K_2\) is a fuzzy relation on \(\underline{M}J_{1}\times \underline{M}J_{2}\).

Similarly, we can show that \(\overline{N}K_1\times \overline{N}K_2\) is a fuzzy relation on \(\overline{M}J_{1}\times \overline{M}J_{2}\). Hence, \(\hat{G}\) is a RFD. \(\square \)

Definition 2.13

Let \(\hat{G}=(\underline{\hat{G}}, \overline{\hat{G}})\) be a RFD. The complement of \(\hat{G}\), denoted by \(\hat{G}^\prime =(\underline{\hat{G}}^\prime , \overline{\hat{G}}^\prime )\) is a RFD, where \(\underline{\hat{G}}^\prime =((\underline{M}J)^\prime ,(\underline{N}K)^\prime )\) and \(\overline{\hat{G}}^\prime =((\overline{M}J)^\prime ,(\overline{N}K)^\prime )\) are FDs such that
  1. (i)

    \((\underline{M}J)^\prime (w) = (\underline{M}J)(w),\) \((\underline{N}K)^\prime (wz) = \min \{(\underline{M}J)(w), (\underline{M}J)(z)\}-(\underline{N}K)(wz),~~\forall ~w,z\in J^*.\)

     
  2. (ii)

    \((\overline{M}J)^\prime (w) = (\overline{M}J)(w),\) \((\overline{N}K)^\prime (wz) = \min \{(\overline{M}J)(w), (\overline{M}J)(z)\}-(\overline{N}K)(wz),~~\forall ~w,z\in J^*.\)

     
Fig. 13

Rough fuzzy digraph \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\)

Example 2.10

Consider a RFD \(\hat{G}\) as shown in Fig. 13.

The complement of \(\hat{G}\) is \(\hat{G}^\prime = (\underline{\hat{G}}^\prime , \overline{\hat{G}}^\prime )\), where \(\underline{\hat{G}}^\prime =((\underline{M}J)^\prime ,(\underline{N}K)^\prime )\) and \(\overline{\hat{G}}^\prime =((\overline{M}J)^\prime ,(\overline{N}K)^\prime )\) are FDs as shown in Fig. 14.

Fig. 14

Rough fuzzy digraph \(\hat{G}^\prime = (\underline{\hat{G}}^\prime , \overline{\hat{G}}^\prime )\)

Definition 2.14

A RFD \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) is self complementary if \(\hat{G}\) and \(\hat{G}^\prime \) are isomorphic, i.e., \(\underline{\hat{G}}\cong \underline{\hat{G}}^\prime \) and \(\overline{\hat{G}}\cong \overline{\hat{G}}^\prime \).

Example 2.11

Let \(J^*=\{t_1,t_2,t_3\}\) be a set and M an ER on \(J^*\) defined by
$$\begin{aligned} \begin{array}{c|@{\quad }c@{\quad }c@{\quad }c} M &{} t_1 &{} t_2 &{} t_3 \\ \hline t_1 &{} 1 &{} 1 &{} 0 \\ t_2 &{} 1 &{} 1 &{} 0 \\ t_3 &{} 0 &{} 0 &{} 1 \\ \end{array} \end{aligned}$$
Let \(J=\{(t_1,0.8),(t_2,0.6),(t_3,0.4)\}\) be a fuzzy set on \(J^*\) and \(MJ=(\underline{M}J,\overline{M}J)\) a RFS, where \(\underline{M}J\) and \(\overline{M}J\) are TLA and TUA of J, respectively, as follows:
$$\begin{aligned} \underline{M}J= & {} \{(t_1,0.6),(t_2,0.6),(t_3,0.4))\}, \\ \overline{M}J= & {} \{(t_1,0.8),(t_2,0.8),(t_3,0.4)\}. \end{aligned}$$
Let \(K^*=\{t_1t_1,t_1t_2,t_1t_3,t_2t_1,t_2t_2,t_2t_3,t_3t_1,t_3t_2,t_3t_3\}\subseteq J^*\times J^*\) and N an ER on \(K^*\) defined by
$$\begin{aligned} \begin{array}{c|@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} N &{} t_1t_1 &{} t_1t_2 &{} t_1t_3 &{} t_2t_1 &{} t_2t_2 &{} t_2t_3 &{} t_3t_1 &{} t_3t_2 &{} t_3t_3 \\ \hline t_1t_1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_1t_2 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_1t_3 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ t_2t_1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_2t_2 &{} 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ t_2t_3 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ t_3t_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 \\ t_3t_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 \\ t_3t_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{array} \end{aligned}$$
Let \(K=\{(t_1t_1,0.4),(t_1t_2,0.3),(t_1t_3,0.2),(t_2t_1,0.3),(t_2t_2,0.4),(qr,0.2),(t_3t_1,0.2),(t_3t_2,0.2),(t_3t_3,0.2)\}\) be a fuzzy set on \(K^*\) and \(NK=(\underline{N}K,\overline{N}K)\) a RFR, where \(\underline{N}K\) and \(\overline{N}K\) are TLA and TUA of K, respectively, as follows:
$$\begin{aligned} \underline{N}K= & {} \{(t_1t_1,0.3),(t_1t_2,0.3),(t_1t_3,0.2),\\&\quad (t_2t_1,0.3),(t_2t_2,0.3),(t_2t_3,0.2),(t_3t_1,0.2),(t_3t_2,0.2),(t_3t_3,0.2)\},\\ \overline{N}K= & {} \{(t_1t_1,0.4),(t_1t_2,0.4),(t_1t_3,0.2),\\&\quad (t_2t_1,0.4),(t_2t_2,0.4),(t_2t_3,0.2),(t_3t_1,0.2),(t_3t_2,0.2),(t_3t_3,0.2)\}. \end{aligned}$$
Thus, \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are FDs as shown in Fig. 15.

The complement of \(\hat{G}\) is \(\hat{G}^\prime = (\underline{\hat{G}}^\prime , \overline{\hat{G}}^\prime )\), where \(\underline{\hat{G}}^\prime =\underline{\hat{G}}\) and \(\overline{\hat{G}}^\prime =\overline{\hat{G}}\) are FDs as shown in Fig. 15 and it can be easily shown that \(\hat{G}\) and \(\hat{G}^\prime \) are isomorphic. Hence, \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\) is a self complementary RFD.

Fig. 15

Self complementary rough fuzzy digraph \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\)

Theorem 2.4

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a self complementary RFD. Then
$$\begin{aligned} \sum \limits _{w,z\in J^*}(\underline{N}K)(wz)=\frac{1}{2}\sum \limits _{w,z\in J^*}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big ) \end{aligned}$$
and
$$\begin{aligned} \sum \limits _{w,z\in J^*}(\overline{N}K)(wz)=\frac{1}{2}\sum \limits _{w,z\in J^*}\big ((\overline{M}J)(w)\wedge (\overline{M}J)(z)\big ). \end{aligned}$$

Proof

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a self complementary RFD. Then there exist two isomorphisms \(\underline{g}:J^*\longrightarrow J^*\) and \(\overline{g}:J^*\longrightarrow J^*,\) respectively, such that
$$\begin{aligned} (\underline{M}J)^\prime (\underline{g}(w))= & {} (\underline{M}J)(w),~\forall ~w\in J^*, \\ (\underline{N}K)^\prime (\underline{g}(w)\underline{g}(z))= & {} (\underline{N}K)(wz),~\forall ~w,z\in J^*,\\ (\overline{M}J)^\prime (\overline{g}(w))= & {} (\overline{M}J)(w),~\forall ~w\in J^*, \\ (\overline{N}K)^\prime (\overline{g}(w)\overline{g}(z))= & {} (\overline{N}K)(wz),~\forall ~w,z\in J^*. \end{aligned}$$
By definition of \(\hat{G}^\prime \), we have
$$\begin{aligned} (\underline{N}K)^\prime (\underline{g}(w)\underline{g}(z))= & {} (\underline{M}J)^\prime (\underline{g}(w))\wedge (\underline{M}J)^\prime (\underline{g}(z))-(\underline{N}K)(\underline{g}(w)\underline{g}(z)) \\ (\underline{N}K)(wz)= & {} (\underline{M}J)(w)\wedge (\underline{M}J)(z)-(\underline{N}K)(\underline{g}(w)\underline{g}(z)) \\ \sum \limits _{w,z\in J^*}(\underline{N}K)(wz)= & {} \sum \limits _{w,z\in J^*}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big )-\sum \limits _{w,z\in J^*}(\underline{N}K)(\underline{g}(w)\underline{g}(z)) \\ \sum \limits _{w,z\in J^*}(\underline{N}K)(wz)= & {} \sum \limits _{w,z\in J^*}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big )-\sum \limits _{w,z\in J^*}(\underline{N}K)(wz) \\ 2\sum \limits _{w,z\in J^*}(\underline{N}K)(wz)= & {} \sum \limits _{w,z\in J^*}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big )\\ \sum \limits _{w,z\in J^*}(\underline{N}K)(wz)= & {} \frac{1}{2}\sum \limits _{w,z\in J^*}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big ) \end{aligned}$$
Similarly, it can be shown that
$$\begin{aligned} \sum \limits _{w,z\in J^*}(\overline{N}K)(wz) = \frac{1}{2}\sum \limits _{w,z\in J^*}\big ((\overline{M}J)(w)\wedge (\overline{M}J)(z)\big ) \end{aligned}$$
This completes the proof. \(\square \)

Theorem 2.5

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD. If
$$\begin{aligned} (\underline{N}K)(wz)=\frac{1}{2}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big ),~\forall ~w,z\in J^*, \end{aligned}$$
and
$$\begin{aligned} (\overline{N}K)(wz)=\frac{1}{2}\big ((\overline{M}J)(w)\wedge (\overline{M}J)(z)\big ),~\forall ~w,z\in J^*, \end{aligned}$$
then \(\hat{G}\) is self complementary.

Proof

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD, where \(\underline{\hat{G}}=(\underline{M}J, \underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J, \overline{N}K)\) are lower and upper approximate FDs, respectively, such that
$$\begin{aligned} (\underline{N}K)(wz)= & {} \frac{1}{2}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big ),~\forall ~w,z\in J^*,\\ (\overline{N}K)(wz)= & {} \frac{1}{2}\big ((\overline{M}J)(w)\wedge (\overline{M}J)(z)\big ),~\forall ~w,z\in J^*. \end{aligned}$$
Then \(\underline{\hat{G}}\approx \underline{\hat{G}}^\prime \) and \(\overline{\hat{G}}\approx \overline{\hat{G}}^\prime \) under the identity map on \(J^*\). \(\square \)

Definition 2.15

Let \(\hat{G}=(\underline{\hat{G}}, \overline{\hat{G}})\) be a RFD. The \(\mu \)-complement of \(\hat{G}\), denoted by \(\hat{G}^\mu =(\underline{\hat{G}}^\mu , \overline{\hat{G}}^\mu )\) is a RFD, where \(\underline{\hat{G}}^\mu =((\underline{M}J)^\mu ,(\underline{N}K)^\mu )\) and \(\overline{\hat{G}}^\mu =((\overline{M}J)^\mu ,(\overline{N}K)^\mu )\) are FDs, respectively, such that
  1. (i)

    \((\underline{M}J)^\mu (w) = (\underline{M}J)(w),~~\forall ~w\in J^*,\) \((\underline{N}K)^\mu (wz) = \left\{ \begin{array}{ll} \min \{(\underline{M}J)(w), (\underline{M}J)(z)\}-(\underline{N}K)(wz), &{} \hbox {if } (\underline{N}K)(wz)>0; \\ 0, &{} \hbox {if }(\underline{N}K)(wz)=0. \end{array} \right. \)

     
  2. (ii)

    \((\overline{M}J)^\mu (w) = (\overline{M}J)(w),~~\forall ~w\in J^*,\) \((\overline{N}K)^\mu (wz) = \left\{ \begin{array}{ll} \min \{(\overline{M}J)(w), (\overline{M}J)(z)\}-(\overline{N}K)(wz), &{} \hbox {if }(\overline{N}K)(wz)>0; \\ 0, &{} \hbox {if }(\overline{N}K)(wz)=0. \end{array} \right. \)

     

Example 2.12

Let \(J^*=\{t_1,t_2,t_3\}\) be a set. Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on \(J^*\), where \(\underline{\hat{G}}=(\underline{M}J, \underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J, \overline{N}K)\) are FDs as shown in Fig. 16.

Then \(\mu \)-complement of \(\hat{G}\) is \(\hat{G}^\mu = (\underline{\hat{G}}^\mu , \overline{\hat{G}}^\mu )\), where \(\underline{\hat{G}}^\mu =((\underline{M}J)^\mu ,(\underline{N}K)^\mu )\) and \(\overline{\hat{G}}^\mu =((\overline{M}J)^\mu ,(\overline{N}K)^\mu )\) are FDs as shown in Fig. 17.

Fig. 16

Rough fuzzy digraph \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\)

Fig. 17

Rough fuzzy digraph \(\hat{G}^\mu = (\underline{\hat{G}}^\mu , \overline{\hat{G}}^\mu )\)

Definition 2.16

A RFD \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) is self\(\mu \)-complementary if \(\hat{G}\) and \(\hat{G}^\mu \) are isomorphic, i.e., \(\underline{\hat{G}}\cong \underline{\hat{G}}^\mu \) and \(\overline{\hat{G}}\cong \overline{\hat{G}}^\mu \).

Example 2.13

Let \(J^*=\{t_1,t_2,t_3,t_4\}\) be a set. Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on \(J^*\), where \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are FDs as shown in Fig. 18.

The \(\mu \)-complement of \(\hat{G}\) is \(\hat{G}^\mu = (\underline{\hat{G}}^\mu , \overline{\hat{G}}^\mu )\), where \(\underline{\hat{G}}^\mu =\underline{\hat{G}}\) and \(\overline{\hat{G}}^\mu =\overline{\hat{G}}\) are FDs as shown in Fig. 18 and it can be easily shown that \(\hat{G}\) and \(\hat{G}^\mu \) are isomorphic. Hence, \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\) is a self \(\mu \)-complementary RFD.

Fig. 18

Self \(\mu \)-complementary rough fuzzy digraph \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\)

Theorem 2.6

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a self \(\mu \)-complementary RFD. Then
$$\begin{aligned} {\mathbb {S}}(\underline{\hat{G}})= & {} \frac{1}{2}\sum \limits _{wz\in (\underline{N}K)^*}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big ),\\ {\mathbb {S}}(\overline{\hat{G}})= & {} \frac{1}{2}\sum \limits _{wz\in (\overline{N}K)^*}\big ((\overline{M}J)(w)\wedge (\overline{M}J)(z)\big ). \end{aligned}$$

Proof

The proof is trivial as it is proved in Theorem 2.4. \(\square \)

Theorem 2.7

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD. If
$$\begin{aligned} (\underline{N}K)(wz)= & {} \frac{1}{2}\big ((\underline{M}J)(w)\wedge (\underline{M}J)(z)\big ),~\forall ~w,z\in J^*,\\ (\overline{N}K)(wz)= & {} \frac{1}{2}\big ((\overline{M}J)(w)\wedge (\overline{M}J)(z)\big ),~\forall ~w,z\in J^* \end{aligned}$$
then \(\hat{G}\) is self \(\mu \)-complementary.

Proof

The proof is trivial as it is proved in Theorem 2.5. \(\square \)

3 Automorphic rough fuzzy digraphs

Definition 3.1

Let \(\hat{G}_1=(\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_2=(\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Then there is a homomorphism\(g:\hat{G}_1\rightarrow \hat{G}_2\) if there are two homomorphisms \(\underline{g}:\underline{\hat{G}}_1\rightarrow \underline{\hat{G}}_2\) and \(\overline{g}:\overline{\hat{G}}_1\rightarrow \overline{\hat{G}}_2\), i.e., there exists a pair of mappings \((\underline{g},\overline{g}):J^*\rightarrow J^*\) such that
  1. (i)

    \((\underline{M}J_1)(w_1)\le (\underline{M}J_2)(\underline{g}(w_1)),\)

    \((\underline{N}K_1)(w_1z_1)\le (\underline{N}K_2)(\underline{g}(w_1)\underline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
  2. (ii)

    \((\overline{M}J_1)(w_1)\le (\overline{M}J_2)(\overline{g}(w_1)),\)

    \((\overline{N}K_1)(w_1z_1)\le (\overline{N}K_2)(\overline{g}(w_1)\overline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
Fig. 19

Rough fuzzy digraph \(\hat{G}_1 =(\underline{\hat{G}}_1, \overline{\hat{G}}_1)\)

Example 3.1

Let \(J^*=\{t_1,t_2,t_3\}\) be a set. Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*\), where \(\underline{\hat{G}}_1=(\underline{M}J_1,\underline{N}K_1)\) and \(\overline{\hat{G}}_1=(\overline{M}J_1,\overline{N}K_1)\) are FDs as shown in Fig. 19.

\(\underline{\hat{G}}_2=(\underline{M}J_2,\underline{N}K_2)\) and \(\overline{\hat{G}}_2=(\overline{M}J_2,\overline{N}K_2)\) are also FDs as shown in Fig. 20.

Thus, \(\hat{G}_1\) and \(\hat{G}_{2}\) are two RFDs. A map \(\underline{g}:J^*\rightarrow J^*\) defined by \(\underline{g}(t_1)=t_1,~\underline{g}(t_2)=t_2\) and \(\underline{g}(t_3)=t_3\). Then we see that:
  • \((\underline{M}J_1)(t_1)<(\underline{M}J_2)(t_1)\), \((\underline{M}J_1)(t_2)<(\underline{M}J_2)(t_2)\), \((\underline{M}J_1)(t_3)<(\underline{M}J_2)(t_3)\),

  • \((\underline{N}K_1)(t_1t_2)<(\underline{N}K_2)(t_1t_2)\), \((\underline{N}K_1)(t_2t_2)<(\underline{N}K_2)(t_2t_2)\), \((\underline{N}K_1)(t_3t_2)<(\underline{N}K_2)(t_3t_2)\).

Hence the map \(\underline{g}\) is a homomorphism. Similarly, the map \(\overline{g}:J^*\rightarrow J^*\) is also a homomorphism.
Fig. 20

Rough fuzzy digraph \(\hat{G}_2 =(\underline{\hat{G}}_2, \overline{\hat{G}}_2)\)

Definition 3.2

Let \(\hat{G}_1=(\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_2=(\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Then there is a isomorphism\(g:\hat{G}_1\rightarrow \hat{G}_2\) if there are two isomorphisms \(\underline{g}:\underline{\hat{G}}_1\rightarrow \underline{\hat{G}}_2\) and \(\overline{g}:\overline{\hat{G}}_1\rightarrow \overline{\hat{G}}_2\), i.e., there exists a pair of bijective mappings \((\underline{g},\overline{g}):J^*\rightarrow J^*\) such that
  1. (i)

    \((\underline{M}J_1)(w_1) = (\underline{M}J_2)(\underline{g}(w_1)),\)

    \((\underline{N}K_1)(w_1z_1) = (\underline{N}K_2)(\underline{g}(w_1)\underline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
  2. (ii)

    \((\overline{M}J_1)(w_1) = (\overline{M}J_2)(\overline{g}(w_1)),\)

    \((\overline{N}K_1)(w_1z_1) = (\overline{N}K_2)(\overline{g}(w_1)\overline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     

Theorem 3.1

Two RFDs are isomorphic if and only if their complements are isomorphic.

Proof

Let \(\hat{G}_1=(\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_2=(\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs and \(\hat{G}_1\cong \hat{G}_2\). Then there exists a pair of bijective mappings \((\underline{g},\overline{g}):J^*\rightarrow J^*\) such that
  1. (i)

    \((\underline{M}J_1)(w_1) = (\underline{M}J_2)(\underline{g}(w_1)),\)

    \((\underline{N}K_1)(w_1z_1) = (\underline{N}K_2)(\underline{g}(w_1)\underline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
  2. (ii)

    \((\overline{M}J_1)(w_1) = (\overline{M}J_2)(\overline{g}(w_1)),\)

    \((\overline{N}K_1)(w_1z_1) = (\overline{N}K_2)(\overline{g}(w_1)\overline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
First we consider (i), by definition of complement:
$$\begin{aligned} (\underline{N}K_1)^\prime (w_1z_1)= & {} \min \{(\underline{M}J_1)(w_1), (\underline{M}J_1)(z_1)\}-(\underline{N}K_1)(w_1z_1)\\= & {} \min \{(\underline{M}J_2)(\underline{g}(w_1)), (\underline{M}J_2)(\underline{g}(z_1))\}-(\underline{N}K_2)(\underline{g}(w_1)\underline{g}(z_1))\\= & {} (\underline{N}K_2)^\prime (\underline{g}(w_1)\underline{g}(z_1)) \end{aligned}$$
Thus, \(\underline{\hat{G}}_{1}^\prime \cong \underline{\hat{G}}_{2}^\prime \). Similarly, \(\overline{\hat{G}}_{1}^\prime \cong \overline{\hat{G}}_{2}^\prime \). Hence, \(\hat{G}_{1}^\prime \cong \hat{G}_{2}^\prime \).

Conversely, suppose that \(\hat{G}_{1}^\prime \cong \hat{G}_{2}^\prime \). Then there exists a pair of bijective mappings

\((\underline{h},\overline{h}):J^*\rightarrow J^*\) such that
  1. (i)
    $$\begin{aligned} (\underline{M}J_1)(w_1)= & {} (\underline{M}J_2)(\underline{h}(w_1)),\nonumber \\ (\underline{N}K_1)^\prime (w_1z_1)= & {} (\underline{N}K_2)^\prime (\underline{h}(w_1)\underline{h}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*. \end{aligned}$$
    (1)
     
  2. (ii)
    $$\begin{aligned} (\overline{M}J_1)(w_1)= & {} (\overline{M}J_2)(\overline{h}(w_1)),\nonumber \\ (\overline{N}K_{1})^\prime (w_1z_1)= & {} (\overline{N}K_2)^\prime (\overline{h}(w_1)\overline{h}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*. \end{aligned}$$
    (2)
     
Using the definition of complement:
$$\begin{aligned} (\underline{N}K_1)^\prime (w_1z_1)= & {} \min \{(\underline{M}J_1)(w_1), (\underline{M}J_1)(z_1)\}\nonumber \\&-(\underline{N}K_1)(w_1z_1),~~\forall ~w_1,z_1\in J^* \end{aligned}$$
(3)
$$\begin{aligned} (\underline{N}K_2)^\prime (\underline{h}(w_1)\underline{h}(z_1))= & {} \min \{(\underline{M}J_2)(\underline{h}(w_1)), (\underline{M}J_2)(\underline{h}(z_1))\}\nonumber \\&-(\underline{N}K_2)(\underline{h}(w_1)\underline{h}(z_1)),~~\forall ~w_1,z_1\in J^* \end{aligned}$$
(4)
Using (3) and (4) in (2) and from (1), we have
$$\begin{aligned} (\underline{N}K_1)(w_1z_1) = (\underline{N}K_2)(\underline{h}(w_1)\underline{h}(z_1)) \end{aligned}$$
(5)
Thus, from (1) and (5), \(\underline{h}:J^*\rightarrow J^*\) is an isomorphism between \(\underline{G}_1\) and \(\underline{G}_2\). Similarly, \(\overline{h}:J^*\rightarrow J^*\) is an isomorphism between \(\overline{G}_1\) and \(\overline{G}_2\).

Hence, \(h:\hat{G}_{1}\rightarrow \hat{G}_{2}\) is an isomorphism, i.e., \(\hat{G}_{1}\cong \hat{G}_{2}\). \(\square \)

Theorem 3.2

If \(G_1\) and \(G_2\) are two isomorphic RFDs then their \(\mu \)- complements \(G_1^\mu \) and \(G_2^\mu \) are also isomorphic.

Proof

The proof is trivial as it is proved in Theorem 3.1. \(\square \)

Definition 3.3

Let \(\hat{G}_1=(\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_2=(\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Then there is a weak isomorphism\(g:\hat{G}_1\rightarrow \hat{G}_2\) if there are two weak isomorphisms \(\underline{g}:\underline{\hat{G}}_1\rightarrow \underline{\hat{G}}_2\) and \(\overline{g}:\overline{\hat{G}}_1\rightarrow \overline{\hat{G}}_2\), i.e., there exists a pair of bijective mappings \((\underline{g},\overline{g}):J^*\rightarrow J^*\) such that
  1. (i)

    \((\underline{M}J_1)(w_1) = (\underline{M}J_2)(\underline{g}(w_1)),\)

    \((\underline{N}K_1)(w_1z_1) \le (\underline{N}K_2)(\underline{g}(w_1)\underline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
  2. (ii)

    \((\overline{M}J_1)(w_1) = (\overline{M}J_2)(\overline{g}(w_1)),\)

    \((\overline{N}K_1)(w_1z_1) \le (\overline{N}K_2)(\overline{g}(w_1)\overline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
Fig. 21

Rough fuzzy digraph \(\hat{G}_1 =(\underline{\hat{G}}_1, \overline{\hat{G}}_1)\)

Example 3.2

Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*=\{t_1,t_2,t_3\}\), where \(\underline{\hat{G}}_{1}=(\underline{M}J_{1}, \underline{N}K_{1})\) and \(\overline{\hat{G}}_{1}=(\overline{M}J_{1}, \overline{N}K_{1})\) are FDs as shown in Fig. 21.

Also \(\underline{\hat{G}}_2=(\underline{M}J_2,\underline{N}K_2)\) and \(\overline{\hat{G}}_2=(\overline{M}J_2,\overline{N}K_2)\) are FDs as shown in Fig. 22.

A map \(\underline{g}:J^*\rightarrow J^*\) defined by \(\underline{g}(t_1)=t_1,~\underline{g}(t_2)=t_2\) and \(\underline{g}(t_3)=t_3\). Then we see that:
  • \((\underline{M}J_1)(t_1)=(\underline{M}J_2)(t_1)\), \((\underline{M}J_1)(t_2)=(\underline{M}J_2)(t_2)\), \((\underline{M}J_1)(t_3)=(\underline{M}J_2)(t_3)\),

  • \((\underline{N}K_1)(t_1t_2)<(\underline{N}K_2)(t_1t_2)\), \((\underline{N}K_1)(t_2t_2)<(\underline{N}K_2)(t_2t_2)\), \((\underline{N}K_1)(t_3t_2)<(\underline{N}K_2)(t_3t_2)\).

Hence the map \(\underline{g}\) is a weak isomorphism. Similarly, the map \(\overline{g}:J^*\rightarrow J^*\) is also a weak isomorphism.
Fig. 22

Rough fuzzy digraph \(\hat{G}_2 =(\underline{\hat{G}}_2, \overline{\hat{G}}_2)\)

Fig. 23

Rough fuzzy digraph \(\hat{G}_1 =(\underline{\hat{G}}_1, \overline{\hat{G}}_1)\)

Definition 3.4

Let \(\hat{G}_1=(\underline{\hat{G}}_1,\overline{\hat{G}}_1)\) and \(\hat{G}_2=(\underline{\hat{G}}_2,\overline{\hat{G}}_2)\) be two RFDs. Then there is a co-weak isomorphism\(g:\hat{G}_1\rightarrow \hat{G}_2\) if there are two co-weak isomorphisms \(\underline{g}:\underline{\hat{G}}_1\rightarrow \underline{\hat{G}}_2\) and \(\overline{g}:\overline{\hat{G}}_1\rightarrow \overline{\hat{G}}_2\), i.e., there exists a pair of bijective mappings \((\underline{g},\overline{g}):J^*\rightarrow J^*\) such that
  1. (i)

    \((\underline{M}J_1)(w_1) \le (\underline{M}J_2)(\underline{g}(w_1)),\)

    \((\underline{N}K_1)(w_1z_1) = (\underline{N}K_2)(\underline{g}(w_1)\underline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     
  2. (ii)

    \((\overline{M}J_1)(w_1) \le (\overline{M}J_2)(\overline{g}(w_1)),\)

    \((\overline{N}K_1)(w_1z_1) = (\overline{N}K_2)(\overline{g}(w_1)\overline{g}(z_1)),~~\forall ~w_1\in J^*,w_1z_1\in K^*.\)

     

Example 3.3

Let \(\hat{G}_{1}=(\underline{\hat{G}}_{1},\overline{\hat{G}}_{1})\) and \(\hat{G}_{2}=(\underline{\hat{G}}_{2},\overline{\hat{G}}_{2})\) be two RFDs on \(J^*=\{t_1,t_2,t_3\}\), where \(\underline{\hat{G}}_{1}=(\underline{M}J_{1}, \underline{N}K_{1})\) and \(\overline{\hat{G}}_{1}=(\overline{M}J_{1}, \overline{N}K_{1})\) are FDs as shown in Fig. 23.

Also \(\underline{\hat{G}}_2=(\underline{M}J_2,\underline{N}K_2)\) and \(\overline{\hat{G}}_2=(\overline{M}J_2,\overline{N}K_2)\) are FDs as shown in Fig. 24.

A map \(\underline{g}:J^*\rightarrow J^*\) defined by \(\underline{g}(t_1)=t_1,~\underline{g}(t_2)=t_2\) and \(\underline{g}(t_3)=t_3\). Then we see that:
  • \((\underline{M}J_1)(t_1)<(\underline{M}J_2)(t_1)\), \((\underline{M}J_1)(t_2)<(\underline{M}J_2)(t_2)\), \((\underline{M}J_1)(t_3)<(\underline{M}J_2)(t_3)\),

  • \((\underline{N}K_1)(t_1t_2)=(\underline{N}K_2)(t_1t_2)\), \((\underline{N}K_1)(t_2t_2)=(\underline{N}K_2)(t_2t_2)\), \((\underline{N}K_1)(t_3t_2)=(\underline{N}K_2)(t_3t_2)\).

Hence the map \(\underline{g}\) is a co-weak isomorphism. Similarly, the map \(\overline{g}:J^*\rightarrow J^*\) is also a co-weak isomorphism.
Fig. 24

Rough fuzzy digraph \(\hat{G}_2 =(\underline{\hat{G}}_2, \overline{\hat{G}}_2)\)

Remark

  • 1. If \(\hat{G}_1=\hat{G}_2=\hat{G}\), then the homomorphism g over itself is called an endomorphism. An isomorphism g over \(\hat{G}\) is called an automorphism.

  • 2. If \(\hat{G}_1=\hat{G}_2\), then the weak and co-weak isomorphisms become isomorphic.

4 Irregular rough fuzzy digraphs

Definition 4.1

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). The indegree of a vertex \(w\in \hat{G}\) is the sum of membership degrees of all edges towards w from other vertices in \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), respectively, denoted by \(id_{\hat{G}}(w)\) and represented by
$$\begin{aligned} id_{\hat{G}}(w)= id_{\underline{\hat{G}}}(w)+id_{\overline{\hat{G}}}(w) \end{aligned}$$
where
$$\begin{aligned} id_{\underline{\hat{G}}}(w)= & {} \sum \limits _{w,z\in (\underline{M}J)^*}zw\\ id_{\overline{\hat{G}}}(w)= & {} \sum \limits _{w,z\in (\overline{M}J)^*}zw \end{aligned}$$
The outdegree of a vertex \(w\in \hat{G}\) is the sum of membership degrees of all edges outward from w to other vertices in \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), respectively, denoted by \(od_{\hat{G}}(w)\) and represented by
$$\begin{aligned} od_{\hat{G}}(w)= od_{\underline{\hat{G}}}(w)+od_{\overline{\hat{G}}}(w) \end{aligned}$$
where
$$\begin{aligned} od_{\underline{\hat{G}}}(w)= & {} \sum \limits _{w,z\in (\underline{M}J)^*}wz\\ od_{\overline{\hat{G}}}(w)= & {} \sum \limits _{w,z\in (\overline{M}J)^*}wz \end{aligned}$$
The pair \(\big (id_{\hat{G}}(w),od_{\hat{G}}(w)\big )\) is called the degree pair of \(w\in \hat{G}\).

Definition 4.2

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). The total degree of a vertex \(w\in \hat{G}\) is the sum of total degrees of w in \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), denoted by, \(td_{\hat{G}}(w)\) and represented by
$$\begin{aligned} td_{\hat{G}}(w)= td_{\underline{\hat{G}}}(w)+td_{\overline{\hat{G}}}(w) \end{aligned}$$
where
$$\begin{aligned} td_{\underline{\hat{G}}}(w)= & {} id_{\underline{\hat{G}}}(w)+od_{\underline{\hat{G}}}(w)+(\underline{M}J)(w) \\ td_{\overline{\hat{G}}}(w)= & {} id_{\overline{\hat{G}}}(w)+od_{\overline{\hat{G}}}(w)+(\overline{M}J)(w) \end{aligned}$$
Fig. 25

Rough fuzzy digraph \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\)

Example 4.1

Consider the RFD \(\hat{G}\) on \(J^*=\{t_1,t_2,t_3,t_4,t_5\}\) as shown in Fig. 25.

From routine calculations, we have
$$\begin{aligned}&id_{\hat{G}}(t_{1})=id_{\underline{\hat{G}}}(t_{1})+id_{\overline{\hat{G}}}(t_{1})=0.3+0.3=0.6, \\&od_{\hat{G}}(t_{1})=od_{\underline{\hat{G}}}(t_{1})+od_{\overline{\hat{G}}}(t_{1})=0.2+0.5=0.7,\\&td_{\hat{G}}(t_{1})=td_{\underline{\hat{G}}}(t_{1})+td_{\overline{\hat{G}}}(t_{1})=1.0+1.9=2.9, \end{aligned}$$
Similarly, we have
$$\begin{aligned}&id_{\hat{G}}(t_{2})=1.7, od_{\hat{G}}(t_{2})=2.0, td_{\hat{G}}(t_{2})=5.4, id_{\hat{G}}(t_{3})=0.8, od_{\hat{G}}(t_{3})=1.3,\\&\quad td_{\hat{G}}(t_{3})=3.3,\\&id_{\hat{G}}(t_{4})=0.6, od_{\hat{G}}(t_{4})=1.0, td_{\hat{G}}(t_{4})=2.8, id_{\hat{G}}(t_{5})=1.9, od_{\hat{G}}(t_{5})=0.6,\\&\quad td_{\hat{G}}(t_{5})=3.3, \end{aligned}$$

Definition 4.3

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). Then \(\hat{G}\) is said to be an irregular RFD if \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are irregular fuzzy digraphs, that is, there is a vertex w in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), which is adjacent to the vertices with distinct degree pairs in \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), respectively.

Example 4.2

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on \(J^*=\{t_1,t_2,t_3,t_4\}\). Thus, \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are FDs as shown in Fig. 26.

From routine calculations, we have
$$\begin{aligned} id_{\underline{\hat{G}}}(t_{1}) = 0.2,&od_{\underline{\hat{G}}}(t_{1}) = 0.5,&id_{\overline{\hat{G}}}(t_{1}) = 0.4,&od_{\overline{\hat{G}}}(t_{1}) = 0.5,\\ id_{\underline{\hat{G}}}(t_{2}) = 0.7,&od_{\underline{\hat{G}}}(t_{2}) = 0.4,&id_{\overline{\hat{G}}}(t_{2}) = 0.7,&od_{\overline{\hat{G}}}(t_{2}) = 0.8,\\ id_{\underline{\hat{G}}}(t_{3}) = 1.0,&od_{\underline{\hat{G}}}(t_{3}) = 0.0,&id_{\overline{\hat{G}}}(t_{3}) = 1.2,&od_{\overline{\hat{G}}}(t_{3}) = 0.0,\\ id_{\underline{\hat{G}}}(t_{4}) = 0.0,&od_{\underline{\hat{G}}}(t_{4}) = 1.0,&id_{\overline{\hat{G}}}(t_{4}) = 0.0,&od_{\overline{\hat{G}}}(t_{4}) = 1.0. \end{aligned}$$
From Fig. 26, it can be seen that \(t_{1}\) is adjacent to \(t_{2}\) and \(t_{3}\) in both fuzzy digraphs \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\). And from above calculations it is also clear that \(t_{2}\) and \(t_{3}\) have distinct degree pairs in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), respectively. Thus, \(\hat{G}\) is an irregular RFD.
Fig. 26

Rough fuzzy digraph \(\hat{G} =(\underline{\hat{G}}, \overline{\hat{G}})\)

Definition 4.4

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). Then \(\hat{G}\) is said to be neighbourly irregular RFD if \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are neighbourly irregular fuzzy digraphs, that is, if every two adjacent vertices in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) have distinct degree pairs, respectively.

Definition 4.5

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). Then \(\hat{G}\) is said to be a highly irregular RFD if \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are highly irregular fuzzy digraphs, that is, if every vertex in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) is adjacent to vertices with distinct degree pairs, respectively.

Proposition 4.1

A highly irregular RFD need not to be a neighbourly irregular RFD.

Fig. 27

Rough fuzzy digraph \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\)

Example 4.3

Consider the RFD \(\hat{G}\) as shown in Fig. 27.

From routine calculations, we have
$$\begin{aligned} id_{\underline{\hat{G}}}(t_{1}) = 0.3,&od_{\underline{\hat{G}}}(t_{1}) = 0.3,&id_{\overline{\hat{G}}}(t_{1}) = 0.3,&od_{\overline{\hat{G}}}(t_{1}) = 0.3,\\ id_{\underline{\hat{G}}}(t_{2}) = 0.4,&od_{\underline{\hat{G}}}(t_{2}) = 0.7,&id_{\overline{\hat{G}}}(t_{2}) = 0.6,&od_{\overline{\hat{G}}}(t_{2}) = 0.9,\\ id_{\underline{\hat{G}}}(t_{3}) = 0.7,&od_{\underline{\hat{G}}}(t_{3}) = 0.4,&id_{\overline{\hat{G}}}(t_{3}) = 0.9,&od_{\overline{\hat{G}}}(t_{3}) = 0.6,\\ id_{\underline{\hat{G}}}(t_{4}) = 0.3,&od_{\underline{\hat{G}}}(t_{4}) = 0.3,&id_{\overline{\hat{G}}}(t_{4}) = 0.3,&od_{\overline{\hat{G}}}(t_{4}) = 0.3. \end{aligned}$$
From Fig. 27, it can be seen that in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), every vertex is adjacent to vertices with distinct degree pairs but vertex \(t_{1}\) is adjacent to \(t_{4}\) having same degree pairs. Therefore, \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) are highly irregular but not neighbourly irregular fuzzy digraphs. Thus, \(\hat{G}\) is a highly irregular RFD but not a neighbourly irregular RFD.

Proposition 4.2

A neighbourly irregular RFD need not to be a highly irregular RFD.

Fig. 28

Rough fuzzy digraph \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\)

Fig. 29

Rough fuzzy digraph \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\)

Example 4.4

Consider the RFD \(\hat{G}\) as shown in Fig. 28.

From routine calculations, we have
$$\begin{aligned} id_{\underline{\hat{G}}}(t_{1}) = 0.40,&od_{\underline{\hat{G}}}(t_{1}) = 0.20,&id_{\overline{\hat{G}}}(t_{1}) = 0.40,&od_{\overline{\hat{G}}}(t_{1}) = 0.30,\\ id_{\underline{\hat{G}}}(t_{2}) = 0.35,&od_{\underline{\hat{G}}}(t_{2}) = 0.53,&id_{\overline{\hat{G}}}(t_{2}) = 0.40,&od_{\overline{\hat{G}}}(t_{2}) = 0.45,\\ id_{\underline{\hat{G}}}(t_{3}) = 0.25,&od_{\underline{\hat{G}}}(t_{3}) = 0.60,&id_{\overline{\hat{G}}}(t_{3}) = 0.25,&od_{\overline{\hat{G}}}(t_{3}) = 0.70,\\ id_{\underline{\hat{G}}}(t_{4}) = 0.10,&od_{\underline{\hat{G}}}(t_{4}) = 0.10,&id_{\overline{\hat{G}}}(t_{4}) = 0.10,&od_{\overline{\hat{G}}}(t_{4}) = 0.10.\\ id_{\underline{\hat{G}}}(t_{5}) = 0.40,&od_{\underline{\hat{G}}}(t_{5}) = 0.20,&id_{\overline{\hat{G}}}(t_{5}) = 0.50,&od_{\overline{\hat{G}}}(t_{5}) = 0.20. \end{aligned}$$
From Fig. 28, it can be easily seen that in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), every two adjacent vertices have distinct degree pairs but vertex \(t_{3}\) is adjacent to \(t_{1}\) and \(t_{5}\), where both \(t_{1}\) and \(t_{5}\) having same degree pairs. Therefore, \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) are neighbourly irregular but not highly irregular fuzzy digraphs. Thus, \(\hat{G}\) is a neighbourly irregular RFD but not a highly irregular RFD.

Definition 4.6

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). Then \(\hat{G}\) is said to be totally irregular RFD if \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are totally irregular fuzzy digraphs, that is, there is a vertex w in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\), which is adjacent to the vertices with distinct total degrees, respectively.

Definition 4.7

Let \(\hat{G}=(\underline{\hat{G}},\overline{\hat{G}})\) be a RFD on a nonempty set \(J^*\). Then \(\hat{G}\) is said to be neighbourly total irregular RFD if \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are neighbourly total irregular fuzzy digraphs, that is, if every two adjacent vertices in both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) have distinct total degrees, respectively.

Example 4.5

Consider the RFD \(\hat{G}\) as shown in Fig. 29.

From routine calculations, we have
$$\begin{aligned} id_{\underline{\hat{G}}}(t_{1}) = 0.1,&od_{\underline{\hat{G}}}(t_{1}) = 1.0,&td_{\underline{\hat{G}}}(t_{1}) = 1.5&~~~&id_{\overline{\hat{G}}}(t_{1}) = 0.2,&od_{\overline{\hat{G}}}(t_{1}) = 1.1,&td_{\overline{\hat{G}}}(t_{1}) = 1.8\\ id_{\underline{\hat{G}}}(t_{2}) = 0.3,&od_{\underline{\hat{G}}}(t_{2}) = 0.7,&td_{\underline{\hat{G}}}(t_{2}) = 1.6&~~~&id_{\overline{\hat{G}}}(t_{2}) = 0.4,&od_{\overline{\hat{G}}}(t_{2}) = 0.8,&td_{\overline{\hat{G}}}(t_{2}) = 1.9\\ id_{\underline{\hat{G}}}(t_{3}) = 0.5,&od_{\underline{\hat{G}}}(t_{3}) = 0.3,&td_{\underline{\hat{G}}}(t_{3}) = 1.2&~~~&id_{\overline{\hat{G}}}(t_{3}) = 0.6,&od_{\overline{\hat{G}}}(t_{3}) = 0.4,&td_{\overline{\hat{G}}}(t_{3}) = 1.5\\ id_{\underline{\hat{G}}}(t_{4}) = 0.9,&od_{\underline{\hat{G}}}(t_{4}) = 0.6,&td_{\underline{\hat{G}}}(t_{4}) = 2.4&~~~&id_{\overline{\hat{G}}}(t_{4}) = 0.9,&od_{\overline{\hat{G}}}(t_{4}) = 0.6,&td_{\overline{\hat{G}}}(t_{4}) = 2.4\\ id_{\underline{\hat{G}}}(t_{5}) = 0.9,&od_{\underline{\hat{G}}}(t_{5}) = 0.1,&td_{\underline{\hat{G}}}(t_{5}) = 1.6&~~~&id_{\overline{\hat{G}}}(t_{5}) = 1.0,&od_{\overline{\hat{G}}}(t_{5}) = 0.2,&td_{\overline{\hat{G}}}(t_{5}) = 1.9 \end{aligned}$$
Thus, both \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) are totally irregular and also neighbourly total irregular fuzzy digraphs. Hence \(\hat{G}\) is totally irregular and neighbourly total irregular RFDs.

Theorem 4.1

Let \(\hat{G}\) be a RFD. Then G is highly irregular fuzzy digraph and neighbourly irregular fuzzy digraph if and only if the degree pairs of all vertices of \(\underline{\hat{G}}\) and \(\overline{\hat{G}}\) are distinct.

Proof

The proof of Theorem 4.1 follows from Theorem 3.13 in [20]. \(\square \)

We state the following propositions without proofs.

Proposition 4.3

If \(\hat{G}\) is a neighbourly irregular RFD. Then \(\hat{G}^\prime \) need not to be a highly irregular RFD.

Proposition 4.4

A neighbourly irregular RFD need not to be a neighbourly total irregular RFD.

Proposition 4.5

A neighbourly total irregular RFD need not to be a neighbourly irregular RFD.

Application

Decision making is the process of making choices by identifying a decision, collecting information, and evaluating all the given alternatives. DM process can help us make more purposeful, thoughtful decisions by systemizing relevant information step-by-step. This approach increases the chances that we will choose the most satisfying alternative among the available alternatives. Traffic congestion has become the major issue of the world. Due to increasing number of vehicles on the roads all over the world, it has become very difficult to move from one place to another. So a person chooses a path having less traffic density in this situation. Here we present an application of DM for a shortest path.

Application for finding the shortest path between any two cities:

Suppose \(J^*=\{C_{1},C_{2},C_{3},C_{4},C_{5}\}\) is a set of cities under consideration. Let M be an ER(where equivalence classes represent the cities of having same road networks) on \(J^*\) defined by
$$\begin{aligned} \begin{array}{c|@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} M &{} C_1 &{} C_2 &{} C_3 &{} C_4 &{} C_5 \\ \hline C_{1} &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 \\ C_{2} &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ C_{3} &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 \\ C_{4} &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 \\ C_{5} &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ \end{array} \end{aligned}$$
Assume that a person Mr. Zain wants to drive from \(C_1\) to \(C_5\). He will select that path which will have the ‘minimum traffic density’ among other paths.
Let \(J=\{(C_{1},0.6),(C_{2},0.7),(C_{3},0.7),(C_{4},0.9),(C_{5},0.8)\}\) be a fuzzy set on \(J^*\) which represents the “traffic density” in each city under consideration and \(MJ=(\underline{M}J,\overline{M}J)\) a RFS, where \(\underline{M}J\) and \(\overline{M}J\) are TLA and TUA approximations of J, respectively, as follows:
$$\begin{aligned} \underline{M}J= & {} \{(C_{1},0.6),(C_{2},0.7),(C_{3},0.6),(C_{4},0.6),(C_{5},0.7)\}, \\ \overline{M}J= & {} \{(C_{1},0.9),(C_{2},0.8),(C_{3},0.9),(C_{4},0.9),(C_{5},0.8)\}. \end{aligned}$$
Let \(K^*=\{C_1C_2,C_1C_3,C_2C_3,C_2C_4,C_2C_5,C_3C_4,C_3C_5,C_4C_3,C_4C_5\}\subseteq J^*\times J^*\) and N an ER on \(K^*\) defined by
$$\begin{aligned} \begin{array}{c|@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} N &{} C_1C_2 &{} C_1C_3 &{} C_2C_3 &{} C_2C_4 &{} C_2C_5 &{} C_3C_4 &{} C_3C_5 &{} C_4C_3 &{} C_4C_5 \\ \hline C_1C_2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 \\ C_1C_3 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ C_2C_3 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ C_2C_4 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ C_2C_5 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ C_3C_4 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ C_3C_5 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 \\ C_4C_3 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ C_4C_5 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1\\ \end{array} \end{aligned}$$
Let \(K=\{(C_1C_2,0.56),(C_1C_3,0.52),(C_2C_3,0.60),(C_2C_4,0.50),(C_2C_5,0.70), (C_3C_4,0.43), (C_3C_5,0.36),(C_4C_3,0.25),(C_4C_5,0.30)\}\) be a fuzzy set on \(K^*\) which represents the “traffic density” between different cities and \(NK=(\underline{N}K,\overline{N}K)\) a RFR, where \(\underline{N}K\) and \(\overline{N}K\) are TLA and TUA approximations of K, respectively, as follows:
$$\begin{aligned} \underline{N}K= & {} \{(C_1C_2,0.30),(C_1C_3,0.25),(C_2C_3,0.50),(C_2C_4,0.50),(C_2C_5,0.70),\\&(C_3C_4,0.25), (C_3C_5,0.30),(C_4C_3,0.25),(C_4C_5,0.30)\}, \\ \overline{N}K= & {} \{(C_1C_2,0.56),(C_1C_3,0.52),(C_2C_3,0.60),(C_2C_4,0.60),(C_2C_5,0.70),\\&(C_3C_4,0.52),(C_3C_5,0.56),(C_4C_3,0.52),(C_4C_5,0.56)\}. \end{aligned}$$
Thus, \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\) and \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\) are FDs as shown in Figs. 30 and 31, respectively.
Fig. 30

Fuzzy digraph \(\underline{\hat{G}}=(\underline{M}J,\underline{N}K)\)

Fig. 31

Fuzzy digraph \(\overline{\hat{G}}=(\overline{M}J,\overline{N}K)\)

By calculations, we have
$$\begin{aligned} \underline{N}K\bullet \overline{N}K= & {} \{(C_1C_2,0.168),(C_1C_3,0.13),(C_2C_3,0.30),(C_2C_4,0.30),(C_2C_5,0.49),\\&(C_3C_4,0.13), (C_3C_5,0.168),(C_4C_3,0.13),(C_4C_5,0.168)\}. \end{aligned}$$
Now the traffic density of each path is shown in Table 1.
Table 1

Traffic densities of the paths

Paths e.g., \(C_1\rightarrow C_2\rightarrow C_3\rightarrow \cdots \rightarrow C_{n-1}\rightarrow C_n\)

Traffic densities of the paths \((\underline{N}K\bullet \overline{N}K)(C_1C_2)+(\underline{N}K\bullet \overline{N}K)(C_2C_3)+\cdots +(\underline{N}K\bullet \overline{N}K)(C_{n-1}C_n)\)

\(C_1\rightarrow C_2\rightarrow C_5\)

0.658

\(C_1\rightarrow C_3\rightarrow C_5\)

0.298

\(C_1\rightarrow C_2\rightarrow C_4\rightarrow C_5\)

0.636

\(C_1\rightarrow C_2\rightarrow C_3\rightarrow C_5\)

0.636

\(C_1\rightarrow C_3\rightarrow C_4\rightarrow C_5\)

0.428

\(C_1\rightarrow C_2\rightarrow C_3\rightarrow C_4\rightarrow C_5\)

0.766

\(C_1\rightarrow C_2\rightarrow C_4\rightarrow C_3\rightarrow C_5\)

0.766

We can see from Table 1, the path \(C_1\rightarrow C_3\rightarrow C_5\) has the minimum traffic density 0.298 among all other paths. So, Mr. Zain will choose this path to drive from \(C_1\) to \(C_5\) (Table 2).

5 Conclusions

RST is an important mathematical tool to deal with uncertainty and vagueness. Fuzzy sets and rough sets are very important models to handle uncertainty from two different perspectives. Present research has shown that these two theories can be combined into a more flexible and expressive framework for modeling and processing incomplete information in information systems. Thus rough fuzzy model is a general framework which is constructed based on crisp approximation space. This model gives more compatibility, precision and flexibility to the system as compared to fuzzy model and rough model. Many authors have constructed various hybrid models consisting of these two models of uncertainty. We have studied one of these innovative hybrid models called rough fuzzy digraph [27] and introduced new methods for its construction. We have defined the complement of RFDs and discussed the properties of self-complementary RFDs. Moreover, we have discussed the concept of irregularity in RFDs, various types of irregularity in RFDs with their properties. Further, we have proposed the new concept of automorphic RFDs which can be viewed as TLA and TUA of automorphic FDs. The concept of RFDs can be applied in various fields of DM. We have applied RFDs to find solution for a shortest path problem.

Notes

Acknowledgements

The authors are very thankful to the Editor and referees for their valuable comments and suggestions for improving the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of the research article.

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Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of the PunjabLahorePakistan

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