Int. J. of Mathematical Sciences and Applications, Vol. 1, No. 3, September 2011 Copyright  Mind Reader Publications www.journalshub.com 1637 A Graph Theoretic Approach: Petri net Sunita Kumawat Department of Applied Mathematics, ASET, Amity university Haryana, Manesar ksunita86@gmail.com skumavat@ggn.amity.edu Abstract This work attempts to understand some of the basic properties of Petri nets and their relationships to directed bipartite graphs. Different forms of directed graphs are widely used in computer science. Normally various names are given to these structures. E.g. directed acyclic graphs (DAGs), control flow graphs (CFGs), task graphs, generalized task graphs (GTGs), state transition diagrams (STDs), state machines, etc. Some structures might exhibit bi- similarity. The justification for this work is that Petri nets are based on graphs and have some similarities to them. Transforming Petri nets into graphs opens up a whole set of new interesting possible experimentations. Normally this is overlooked. Directed Graphs have a lot of theory and research associated with them. This work could be further developed and used for Petri net evaluation. The related works justifies the reasoning how and why Petri nets are obtained or supported using graphs. The transformation approach can be formal or informal. The main problem tackled is how graphs can be obtained from Petri nets. Possible solutions that use reduction methods to simplify the Petri net are presented. Different methods to extract graphs from the basic or fundamental Petri net classes are explained. Some examples are given and the findings are briefly discussed. Keywords: Graphs Petri Nets, Transformation, Reduction INTRODUCTION Petri Nets are expressive graphical formalisms that serve to model discrete event behavior that takes place in different systems [12]-[15]. They are designed to model system behavior like: sequential behavior, concurrency, mutual exclusion, non-determinism, choice and conflict. Petri nets are classified into different classes ranging from elementary nets to higher order nets, colored Petri nets and object oriented nets. All these classes can be converted to time Petri nets. Ordinary Petri nets have a 'dual identity' they can be represented graphically or by using equations. These can be analyzed using mathematical models. Petri nets have at least three decades of use. Normally speaking, the analysis of Petri nets is based on i) structural properties and ii) behavioral properties [6]. The structural properties of Petri nets are suitable to understand the basic underlying structure. If the Petri net is viewed, basic structural features can be seen. E.g. the Petri net can be cycle free (acyclical) [9]. It could have bounded places, etc. On the other hand behavioral properties explain the behavior of the Petri net. These properties cannot be applied to all types of Petri nets especially if the net is unbounded or improperly designed. Some basic behavioral properties are i) reachability, ii) boundedness, iii) safeness, iv) conservativeness, v) liveness, vi) reversiblility, vii) repetitiveness, viii) home states. One of the salient points for using Petri nets is precisely the ability to transform them or obtain them from other formalisms or notations. Petri nets are classified as directed bi-partite graphs, definitely sharing some common properties with graphs. This means that they could be transformed into graphs and analyzed from this point of view. The work in this paper is restricted to the basic or fundamental classes of Petri nets. PRELIMINARIES A normal Petri net is basically defined as directed bipartite graph or bipartite digraph that can be basically represented as a five tuple A place-transition net (PT-Net) is a quadruplet PN = P, T, F, W, M 0 , where P = {p 1 , p 2 . . . p m } is the set of places, T = {t 1 , t 2 , . . . , t m } is the set of transitions, such that P  T  and P  T= , F (P × T)  (T × P) is the set of arcs and W: F {1, 2 ...} is the weight function. M 0 represents the initial marking PROBLEM FORMULATION