scholarly journals Rough sets in graphs using similarity relations

2022 ◽  
Vol 7 (4) ◽  
pp. 5790-5807
Author(s):  
Imran Javaid ◽  
◽  
Shahroz Ali ◽  
Shahid Ur Rehman ◽  
Aqsa Shah
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Sufficient Conditions ◽  
Membership Functions ◽  
Similarity Relations ◽  
Complete Bipartite Graphs ◽  
Essential Set ◽  
Essential Sets ◽  
Necessary And Sufficient ◽  
Clustering Criterion

<abstract><p>In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.</p></abstract>

scholarly journals On rough sets induced by fuzzy relations approach in semigroups

2018 ◽  
Vol 16 (1) ◽  
pp. 1634-1650
Author(s):  
Rukchart Prasertpong ◽  
Manoj Siripitukdet
Keyword(s):  
Prime Ideal ◽  
Rough Set ◽  
Rough Sets ◽  
Sufficient Conditions ◽  
Unit Interval ◽  
Fuzzy Relation ◽  
Prime Ideals ◽  
Fuzzy Relations ◽  
Completely Prime Ideal

AbstractIn this paper, we introduce a rough set in a universal set based on cores of successor classes with respect to level in a closed unit interval under a fuzzy relation, and some interesting properties are investigated. Based on this point, we propose a rough completely prime ideal in a semigroup structure under a compatible preorder fuzzy relation, including the rough semigroup and rough ideal. Then we provide sufficient conditions for them. Finally, the relationships between rough completely prime ideals (rough semigroups and rough ideals) and their homomorphic images are verified.

scholarly journals Note on group distance magic complete bipartite graphs

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Sylwia Cichacz
Keyword(s):  
Abelian Group ◽  
Complete Graph ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Distance Labeling ◽  
Distance Magic Labeling ◽  
Odd Order ◽  
Complete Bipartite Graphs ◽  
Necessary And Sufficient ◽  
Magic Constant

AbstractA Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|.In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).

scholarly journals Packings and Coverings of Various Complete Graphs with the 4-Cycle with a Pendant Edge

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Brandon Coker ◽  
Gary D. Coker ◽  
Robert Gardner ◽  
Yan Xia
Keyword(s):  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Necessary And Sufficient Conditions ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Necessary And Sufficient

We consider the packings and coverings of complete graphs with isomorphic copies of the 4-cycle with a pendant edge. Necessary and sufficient conditions are given for such structures for (1) complete graphs , (2) complete bipartite graphs , and (3) complete graphs with a hole . In the last two cases, we address both restricted and unrestricted coverings.

scholarly journals Fuzzy BCI-subalgebras with interval-valued membership functions

2001 ◽  
Vol 25 (2) ◽  
pp. 135-143
Author(s):  
Sung Min Hong ◽  
Young Bae Jun ◽  
Seon Jeong Kim ◽  
Gwang Il Kim
Keyword(s):  
Fuzzy Set ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Membership Functions ◽  
Necessary And Sufficient ◽  
Interval Valued

The purpose of this paper is to define the notion of an interval-valued fuzzy BCI-subalgebra (briefly, an i-v fuzzy BCI-subalgebra) of a BCI-algebra. Necessary and sufficient conditions for an i-v fuzzy set to be an i-v fuzzy BCI-subalgebra are stated. A way to make a new i-v fuzzy BCI-subalgebra from old one is given. The images and inverse images of i-v fuzzy BCI-subalgebras are defined, and how the images or inverse images of i-v fuzzy BCI-subalgebras become i-v fuzzy BCI-subalgebras is studied.

scholarly journals Graph approximation on similarity based rough sets

2020 ◽  
Vol 15 (2) ◽  
pp. 25-36
Author(s):  
Dávid Nagy ◽  
Tamás Mihálydeák ◽  
László Aszalós
Keyword(s):  
Data Mining ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Correlation Clustering ◽  
The Real ◽  
Similarity Relations ◽  
Distinguished Member

Abstract:Correlation clustering is a widely used technique in data mining. The clusters contain objects, which are typically similar to one another and different from objects from other groups. In the authors previous works the possible usage of correlation in rough set theory were investigated. In rough set theory, two objects are treated as indiscernible if all of their attribute values are the same. A base set contains those objects that are indiscernible from one another. The partition, gained from the correlation clustering, can be understood as the system of base sets, as the clusters contain the typically similar objects (not just to a distinguished member) and it considers the real similarity among the objects. In this work the extension of this study is presented, using the method to approximate graphs representing similarity relations.

scholarly journals Continuous Monotonic Decomposition of Some Standard Graphs by using an Algorithm

2019 ◽  
Vol 2 (8) ◽  
pp. 6-9
Keyword(s):  
Tensor Product ◽  
Bipartite Graph ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Necessary And Sufficient Conditions ◽  
Research Article ◽  
Complete Bipartite Graphs ◽  
Disjoint Sets ◽  
Necessary And Sufficient ◽  
Tripartite Graphs

In this paper we elaborate an algorithm to compute the necessary and sufficient conditions for the continuous monotonic star decomposition of the bipartite graph Km,r and the number of vertices and the number of disjoint sets. Also an algorithm to find the tensor product of Pn  Ps has continuous monotonic path decomposition. Finally we conclude that in this paper the results described above are complete bipartite graphs that accept Continuous monotonic star decomposition. There are many other classes of complete tripartite graphs that accept Continuous monotonic star decomposition. In this research article Extended to complete m-partite graphs for grater values of m. Also the algorithm can be developed for the tensor product of different classes such as Cn Wn K1,n , , with Pn

A New Description of Transversal Matroids Through Rough Set Approach

2021 ◽  
Vol 179 (4) ◽  
pp. 399-416
Author(s):  
Zhaohao Wang
Keyword(s):  
Rough Set ◽  
Rough Set Theory ◽  
Sufficient Conditions ◽  
Mining Machine ◽  
Matroid Theory ◽  
Approximation Number ◽  
Upper Approximation ◽  
Transversal Matroid ◽  
Necessary And Sufficient ◽  
The Relationship

Matroid theory is a useful tool for the combinatorial optimization issue in data mining, machine learning and knowledge discovery. Recently, combining matroid theory with rough sets is becoming interesting. In this paper, rough set approaches are used to investigate an important class of matroids, transversal matroids. We first extend the concept of upper approximation number functions in rough set theory and propose the notion of generalized upper approximation number functions on a set system. By means of the new notion, we give some necessary and sufficient conditions for a subset to be a partial transversal of a set system. Furthermore, we obtain a new description of a transversal matroid by the generalized upper approximation number function. We show that a transversal matroid can be induced by the generalized upper approximation number function which can be decomposed into the sum of some elementary generalized upper approximation number functions. Conversely, we also prove that a generalized upper approximation number function can induce a transversal matroid. Finally, we apply the generalized upper approximation number function to study the relationship among transversal matroids.

Interpretations of belief functions in approximation operators by covering

2021 ◽  
pp. 1-11
Author(s):  
Yan-Lan Zhang ◽  
Chang-Qing Li
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Evidence Theory ◽  
Sufficient Conditions ◽  
Belief Functions ◽  
Belief Structure ◽  
Approximation Operators ◽  
Necessary And Sufficient ◽  
Covering Rough Set

The rough set theory and the evidence theory are two important methods used to deal with uncertainty. The relationships between the rough set theory and the evidence theory have been discussed. In covering rough set theory, several pairs of covering approximation operators are characterized by belief and plausibility functions. The purpose of this paper is to review and examine interpretations of belief functions in covering approximation operators. Firstly, properties of the belief structures induced by two pairs of covering approximation operators are presented. Then, for a belief structure with the properties, there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions induced by one of the two pairs of covering approximation operators. Moreover, two necessary and sufficient conditions for a belief structure to be the belief structure induced by one of the two pairs of covering approximation operators are presented.

MONOTONE APPROXIMATION OF AGGREGATION OPERATORS USING LEAST SQUARES SPLINES

2002 ◽  
Vol 10 (06) ◽  
pp. 659-676 ◽  
Author(s):  
G. BELIAKOV
Keyword(s):  
Least Squares ◽  
Fuzzy Systems ◽  
Sufficient Conditions ◽  
Aggregation Operators ◽  
Linear Functions ◽  
Membership Functions ◽  
Polynomial Splines ◽  
Monotone Approximation ◽  
Necessary And Sufficient ◽  
Linear Restrictions

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

scholarly journals INCREMENTALLY UPDATING APPROXIMATION IN INCOMPLETE INFORMATION SYSTEMS UNDER THE VARIATION OF OBJECTS

2020 ◽  
Vol 36 (4) ◽  
pp. 365-379
Author(s):  
Tran Thi Thanh Huyen ◽  
Le Ba Dung ◽  
Nguyen Do Van ◽  
Mai Van Dinh
Keyword(s):  
Information System ◽  
Information Systems ◽  
Rough Set ◽  
Rough Sets ◽  
Membership Functions ◽  
Approximation Space ◽  
The Third ◽  
Incomplete Information Systems ◽  
The Relationship ◽  
Over Time

Rough membership functions in covering approximation space not only give numerical characterizations of covering-based rough set approximations, but also establish the relationship between covering-based rough sets and fuzzy covering-based rough sets. In this paper, we introduce a new method to update the approximation sets with rough membership functions in covering approximation space. Firstly, we present the third types of rough membership functions and study their properties. And then, we consider the change of them while simultaneously adding and removing objects in the information system. Based on that change, we propose a method for updating the approximation sets when the objects vary over time.

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