International Journal of Mathematics and Mathematical Sciences

Volume 2018, Article ID 6045358, 11 pages

https://doi.org/10.1155/2018/6045358

## Metric Induced Morphological Operators on Intuitionistic Fuzzy Hypergraphs

^{1}Department of Computer Applications, Cochin University of Science and Technology, Kochi, India^{2}Department of CS, RSET, Kakkanad, Kochi, India^{3}Department of Computer Applications, CUSAT, Kochi, India^{4}Department of Basic Sciences and Humanities, Rajagiri School of Engineering and Technology, Kochi, India

Correspondence should be addressed to P. M. Dhanya; moc.liamg@skhsejar.aynahd

Received 9 February 2018; Accepted 4 June 2018; Published 4 July 2018

Academic Editor: Henryk Hudzik

Copyright © 2018 P. M. Dhanya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A hypergraph consisting of hyperedges and nodes can be made intuitionistic fuzzy hypergraph (IFHG) by assigning membership and nonmembership degrees for both nodes and edges. Just as a hypergraph, an IFHG is also having hyperedges consisting of many nodes. Many flavours of a given IFHG can be created by applying morphological operators like dilation, erosion, adjunction, etc. The focus of this paper is to define morphological adjunction, opening, closing, and Alternative Sequential Filter (ASF) on IFHG. The system modeled in this way finds application in text processing, image processing, network analysis, and many other areas.

#### 1. Introduction

Let be an intuitionistic fuzzy hypergraph with nodes and hyperedges . is the set of nodes with membership degree and non membership degree . Depending on the membership degree , the node can be treated as high priority, medium priority, and low priority. The non membership degree . The sum of the membership degree and non membership degree of the node is less than or equal to 1 [1], i.e., . is the set of hyperedges with membership degree and non membership degree . If all the nodes in a hyperedge have , then is the supremum of all in that edge. In such a case . If there is at least one node with , then of that edge is the supremum of all in that edge. In such a case . If all the nodes of an edge are having , then its = = 0.5. The sum of the membership degree and non membership degree of the hyperedge is less than or equal to 1 [1], i.e., . Depending on the membership degree , the hyperedge can be treated as high priority, medium priority, and low priority. Since can take any value from 0.0 to 1.0, priorities are not limited to the above three. Also range priorities can be set for nodes and edges. Morphological operations like dilation, erosion, adjunction are already defined on hypergraphs [2]. Dilation operation [3] and erosion [4] is already defined on IFHG. The purpose of this paper is to define adjunction, opening, closing, and ASF on IFHG.

#### 2. Related Works

Many morphological operators [5] like dilation, erosion, adjunction, and duality were applied on lattice of all subgraphs, where these are further applied in binary and grey scale image denoising. The authors have designed filters for which graph is both the input and the output. Various edge-vertex correspondences [6], edge-vertex adjunctions, vertex dilation, vertex erosion, edge dilation, edge erosion, graph dilation, graph erosion, opening, closing filters, half opening, and half closing on graphs, are illustrated with examples which finds application in image filtering. Images were represented using set union [7] of hyperedges and were subjected to contraharmonic mean filter for salt and pepper noise removal. The method gave better results in terms of visual quality, peak signal to noise ratio, and mean absolute error. A framework was proposed to build morphological operators for analyzing and filtering objects defined on simplicial complex spaces [8] where a simplex is any finite nonempty set. This finds application in mesh and image filtering. Applications of mathematical morphology to weighted graphs [9] were also developed. By converting the image to a tree representation, the authors have shown how to filter the image using connected filters. Mathematical morphology [10], which is the basis of morphological image processing, finds application in the field of digital image processing, area of graphs, surface mesh, solids, and other spatial structures.

Now coming to the field of hypergraphs, lattice structures on hypergraphs introduced in [11] have shown many properties like partial ordering, infimum, supremum, isomorphism, etc. The authors also introduced complete lattice, dualities, discrete probability distribution on vertices and hypergraph similarity based on dilation. Mathematical morphology of hypergraphs were also used for classification or matching problems [12] on data represented by hypergraphs. As an example, the authors have applied it on a 2D image and they proposed further applications of hypergraph in image analysis. New similarity measures and pseudometrics on lattices of hypergraphs are detailed in [13] which can be incorporated in existing system for hypergraph-based feature selection, indexing, retrieval, and matching.

Intuitionistic fuzzy graphs [14] are used for clustering with the help of many operations like complement, join, union, intersection, ringsum, Cartesian product, composition, etc. The concepts like cut, edge strength, incidence matrix of an IFHG [1] are also used in clustering. Isomorphism between two IFHG and the Cartesian product of two IFS over the same universe are detailed in [15], where they also illustrated indegree and outdegree of vertex , weak isomorphism, and coweak isomorphism. A hypernetwork [16] can be created with processors as vertices and connections between the processors modeled as hyperedge. Radio coverage networks in a geographic region can be modeled with radio receivers as vertices, where the membership values signify the quality of reception of a station/radio. The authors also proposed further research in intuitionistic fuzzy soft hypergraphs and rough hypergraphs. An application with intuitionistic fuzzy sets for career choice [17] which is a decision-making system was developed where the system represented the performance of students using membership, nonmembership, and hesitation margin. They applied normalized Euclidian distance to determine the apt career choice. Operations on transversals of IFDHG, their union, intersection, addition, structural subtraction, multiplication, and complement are defined and discussed in [18]. The authors also propose to work on application in coloring of IFHG. Generalized strong IFHG (GSIFHG), spanning IFHG, and generalized strong spanning IFHG were discussed in [19], which can be used to analyze the structure of a system and to represent a partition, covering, and clustering.

#### 3. Preliminary Definitions

Let us define , where and as given in Figure 1. Here nodes with low priority are having , nodes of medium priority are having , and nodes of high priority are with . Let be obtained by cut on . Let be obtained by cut on . Here corresponds to membership degree and corresponds to nonmembership degree. The details of the IFHGs , , and are given in Tables 1, 2, and 3, respectively.