# On Soft Graphs and Chained Soft Graphs

K. P. Ratheesh
DOI: 10.4018/IJFSA.2018040105
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## Abstract

Soft set theory has a rich potential for application in many scientific areas such as medical science, engineering and computer science. This theory can deal uncertainties in nature by parametrization process. In this article, the authors explore the concepts of soft relation on a soft set, soft equivalence relation on a soft set, soft graphs using soft relation, vertex chained soft graphs and edge chained soft graphs and investigate various types of operations on soft graphs such as union, join and complement. Also, it is established that every fuzzy graph is an edge chained soft graph.
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## 1. Introduction

In the year 1999, Russian mathematician D. Molodtsov initiated the theory of soft sets. His theory was primarily based on parametrized family of sets. Before the development of soft set theory, fuzzy set theory was used to handle uncertain situations widely. But the major difficulty with fuzzy sets is to set a suitable membership function for a particular problem. In his remarkable paper, D. Molodtsov (1999) proposed soft sets as a new mathematical tool for dealing with uncertainties and which is free from difficulties faced by previous fuzzy set theory.

In order to solidify the theory of soft set, Maji et al. (2002), defined some basic terms of the theory such as soft subset, equality of two soft sets, complement soft set of a soft set, null soft set and absolute soft set. Also, they defined binary operations such as AND, OR, union and intersection and verified De Morgan’s laws in soft set theory context. Ali et al. (2009), introduced some new definitions such as the restricted union, the restricted intersection, the restricted difference and the extended intersection of two soft sets and also established that certain De Morgan’s law hold in soft set theory. In addition to this, Sezgin et al. (2011) presented restricted symmetric difference of soft sets and explore its properties.

Babitha et al. (2010) introduced the concept of soft set relation and functions from the fact of cartesian product and presented many related concepts. Yang et al. (2011) studied the concept of the anti-reflexive kernel, symmetric kernel, reflexive closure and symmetric closure of a soft set relation. Finally, soft set relation mapping and inverse soft set relation mappings were proposed and some related properties were discussed. Zhou et al (2015) defined the concepts Z-soft set relations, Z-anti reflexive interior operator and Z-anti reflexive closure operator and the algebraic and topological characterizations of Z-soft set relations are studied.

Qin et al. (2014) proposed the notion of soft relation which is a generalization of the notion of soft set relation and some related properties are examined. Also, Kharal et al. (2011) defined mapping on soft classes and studied several properties of images and inverse images of soft sets supported by examples and counterexamples. These notions were applied to the problem of medical diagnosis in medical expert systems. Çağman et al. (2010), defined soft matrices and their operations to construct a soft max-min decision making method which can be successfully applied to the problems that contain uncertainties.

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