## Abstract

This article investigates the notion of a minimum spanning tree (MST) of an undirected type-2 fuzzy weighted connected graph (UT2FWCG), where the edge weights are represented as discrete type-2 fuzzy variables. A modified type-2 fuzzy Borüvka’s algorithm (MT2FBA) is proposed to determine an MST of UT2FWCG. The effective weight of the MST is calculated, and its defuzzified value is compared with an equivalent MST with deterministic weights. The working principle of the algorithm includes ranking, addition operation and defuzzification of discrete type-2 fuzzy variables. The defuzzified effective weight of the MST is determined using the critical value (CV)-based type reduction technique and centroid method. The proposed algorithm is numerically illustrated with suitable examples. Moreover, the simulation results of nine random instances of UT2FWCG are also studied to analyze the proposed MT2FBA properly.

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## Acknowledgements

The authors would like to express their gratitude and indebtedness to the Editor-in-Chief, Managing Editor, and the anonymous referees for their careful reading and valuable comments.

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## Appendices

### Appendix

We present some fundamental concepts of T1FS in this section.

### A.1 Type-1 fuzzy variable

Zadeh (1975) introduced the possibility theory to deal with the uncertainty involved in the occurrence of an element in a fuzzy set. The membership function \({\mu }_{\tilde{A }}(x)\) of an element \(x\) in a fuzzy set \(\tilde{A }\) is then termed as degree of possibility that the element belongs to the set. In other words, the possibility of occurrence of an element \(x\) in \(\tilde{A }\) is characterized by a membership function \({\mu }_{\tilde{A }}(x)\), where \(x\in X\) and \({\mu }_{\tilde{A }}\left(x\right)\in \left[\mathrm{0,1}\right]\).

A type-1 fuzzy variable (T1FV) is defined as a function from the possibility space (\(U\), \(\mathcal{A}\),\(Pos\)) to the set of real numbers **ℜ** to describe fuzzy phenomena, where \(U\) is the non-empty set of points, \(\mathcal{A}\) is power set of \(U\). Furthermore, for a fuzzy event \(\left\{\tilde{\gamma }\in P\right\}, P\subset \) ℜ, the possibility measure \(\left(Pos\right)\) (Zadeh 1978), necessity measure \(\left(Nec\right)\) and credibility measure \(\left(Cr\right)\) (Liu and Liu 2002) are, respectively, defined in Eqs. (A1–A3) as follows.

and

### A.2 Fuzzy possibility measure

(Liu and Liu 2007) Let \(\widetilde{Pos }:\) \(\mathcal{A}\)↦\(\mathfrak{R}\left(\left[0,1\right]\right)\) be a set of functions defined on \(\mathcal{A}\) such that \(\{\tilde{{Pos}}({\varvec{A}})|\) \(\mathcal{A}\) \(\ni A \mathrm{atom}\}\) is a collection of mutually independent RFVs. \(\widetilde{Pos }\) is said to be the fuzzy possibility measure if it satisfies the following conditions: (i) \(\tilde{{Pos}}\left(\varnothing \right)=\tilde{0 }\) (ii) For any subclass \(\left\{{A}_{i}|i\in {\mathbb{I}}\right\}\mathrm{ of}\) \(\mathcal{A}\) (finite, countable or uncountable), \(\widetilde {{{~Pos}}}\left( {\bigcup\nolimits_{i \in } {{A_i}} } \right) = \begin{array}{*{20}{c}}{{{sup}}}\\{i \in }\end{array}\widetilde {{{Pos}}}\left( {{A_i}} \right)\). Then, the triplet \((U,\) \(\mathcal{A}\)\(,\widetilde{Pos}\)) is called fuzzy possibility space.

### Remark A.1

Let \(U\) be the universe of discourse and \(\mathcal{A}\) the power set of \(U\); then, an atoms of \(\mathcal{A}\) are all single-point sets \(\left\{\delta \right\}, \delta \in U.\)

### Remark A.2

(Liu and Liu 2007) If \({\mu }_{\tilde{{Pos}}\left(U\right)}\left(1\right)=1\) then \(\widetilde{Pos}\) becomes a regular fuzzy possibility measure, which is the generalization of scalar possibility measure, i.e., if \(B\in \)\(\mathcal{A}\) , then \(\widetilde{Pos}(B)\) is a crisp value within the interval \(\left[\mathrm{0,1}\right]\) instead of an RF, then \(\widetilde{Pos}\) is just a possibility measure.

### Remark A.3

(Liu and Liu 2007) If \(U\) is a finite set, then \(\widetilde {{{Pos}}}\left( {\bigcup\nolimits_{i \in } {{A_i}} } \right) = \begin{array}{*{20}{c}}{{{sup}}}\\{1 \le i \le n}\end{array}\widetilde {{{Pos}}}\left( {{A_i}} \right)\) for any finite subclass \(\left\{{A}_{i}, i=\mathrm{1,2},\ldots ,n\right\}\) of \(\mathcal{A}\)

### Centroid method

(Yager 1981) Centroid method is one of the defuzzification techniques of TIFVs. This method is used to obtain the center of area for continuous and discrete T1FVs, which are defined, respectively, in (A4) and (A5).

and

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Dan, S., Majumder, S., Kar, M.B. *et al.* On type-2 fuzzy weighted minimum spanning tree.
*Soft Comput* **25, **14873–14892 (2021). https://doi.org/10.1007/s00500-021-06052-1

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### Keywords

- Minimum spanning tree
- Type-2 fuzzy variable
- Satisfaction function
- CV-based type reduction
- Defuzzification
- Borüvka’s algorithm