On Computing Domination Set in Intuitionistic Fuzzy Graph

Abstract

In this paper, the concept of minimal intuitionistic dominating vertex subset of an intuitionistic fuzzy graph was considered, and on its basis, the notion of a domination set as an invariant of the intuitionistic fuzzy graph was introduced. A method and an algorithm for finding all minimal intuitionistic dominating vertex subset and domination set was proposed. This method is the generalization of Maghout's method for fuzzy graphs. The example of finding the domination set of the intuitionistic fuzzy graph were considered as well.

Domination of Bipolar Fuzzy Graphs in Various Settings

“…The main reason for this phenomenon is that the definition of minimal bipolar intuitionistic dominating vertex subset is too strict, and the deeper reason in the definition of 𝛽(X 1 ) < 𝛽(X 2 ) is too harsh. Recall that in Bozhenyuk et al [23], for (X 1 ) = ( B (v 1 , v � 1 ), B (v 1 , v � 1 )) and (X 2 ) = ( B (v 2 , v � 2 ), B (v 2 , v � 2 )) , 𝛽(X 1 ) < 𝛽(X 2 ) i m p l i e s 𝜇 B (v…”

“…For intuitionistic fuzzy graph, Bozhenyuk et al [23] determined that the intuitionistic dominating sets have property (0, 1, ) ≤ 0 1 ≤ 0 2 ≤ ⋯ ≤ 0 n ≤ (1, 0) . When it comes to bipolar setting, if we expand it directly, then it will become (0, 0, 1, −1) ≤ 0 1 ≤ 0 2 ≤ ⋯ ≤ 0 n ≤ (1, −1, 0, 0) .…”

“…However, it is obvious that for bipolar intuitionistic fuzzy graph and bipolar intuitionistic dominating set, this property is not hold any more. In Bozhenyuk et al [23], the authors gave more discussions, algorithm and numerical examples on how to compute the domination set in intuitionistic fuzzy graph, and unfortunately, these statement can be extended to bipolar setting.…”

“…Since bipolar fuzzy sets can be used to represent the uncertain attributes of both positive and negative sides of things, they are widely used in fuzzy decision-making systems. To structurally express the fuzzy attributes with bipolar features, a bipolar fuzzy graph is defined, and the concepts of path, connectivity, and topological indices of the fuzzy graph are also defined successively (for more details on graph, fuzzy graph in various settings, see Santiago et al [21], Ali et al [22], Bozhenyuk et al [23], Das et al [24], Akram et al [25], Das et al [26], Kalathian et al [27], Gao et al [28] and [29], and Gong and Hua [30]). Although the bipolar fuzzy graph framework is successfully applied in some fields, its fuzzy characteristics and related attributes still need to be studied in most settings.…”

An Application of Product of Intuitionistic Fuzzy Incidence Graphs in Textile Industry

“…Sahoo et al [24] initiated new ideas in IFGs with application in the water supply system. Bozhenyuk et al [25] presented an idea of minimal intuitionistic dominating vertex subset of an IFG.…”