Bipolar fuzzy graphs Muhammad Akram Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore 54000, Pakistan article info Article history: Received 28 May 2010 Received in revised form 10 June 2011 Accepted 21 July 2011 Available online 29 July 2011 Keywords: Bipolar fuzzy graphs Isomorphisms Strong bipolar fuzzy graphs Self complementary abstract In this paper, we introduce the notion of bipolar fuzzy graphs, describe various methods of their construction, discuss the concept of isomorphisms of these graphs, and investigate some of their important properties. We then introduce the notion of strong bipolar fuzzy graphs and study some of their properties. We also discuss some propositions of self com- plementary and self weak complementary strong bipolar fuzzy graphs. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction In 1965, Zadeh [51] introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multi- agent systems, pattern recognition, robotics, computer networks, expert systems, decision making and automata theory. In 1994, Zhang [56,57] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. Bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is [1, 1]. In a bipolar fuzzy set, the membership degree 0 of an ele- ment means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indi- cates that the element somewhat satisfies the property, and the membership degree [1, 0) of an element indicates that the element somewhat satisfies the implicit counter-property. Although bipolar fuzzy sets and intuitionistic fuzzy sets look sim- ilar to each other, they are essentially different sets [27]. In many domains, it is important to be able to deal with bipolar infor- mation. It is noted that positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. This domain has recently motivated new research in several directions. In particular, fuz- zy and possibilistic formalisms for bipolar information have been proposed [18], because when we deal with spatial informa- tion in image processing or in spatial reasoning applications, this bipolarity also occurs. For instance, when we assess the position of an object in a space, we may have positive information expressed as a set of possible places and negative informa- tion expressed as a set of impossible places. As another example, let us consider the spatial relations. Human beings consider ‘‘left’’ and ‘‘right’’ as opposite directions. But this does not mean that one of them is the negation of the other. The semantics of ‘‘opposite’’ captures a notion of symmetry rather than a strict complementation. In particular, there may be positions which are considered neither to the right nor to the left of some reference object, thus leaving some room for indetermination. This corresponds to the idea that the union of positive and negative information does not cover the whole space. In 1975, Rosenfeld [45] introduced the concept of fuzzy graphs. The fuzzy relations between fuzzy sets were also consid- ered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. Later on, Bhattacharya [8] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.07.037 E-mail addresses: makrammath@yahoo.com, m.akram@pucit.edu.pk Information Sciences 181 (2011) 5548–5564 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins