Remarks on the Reachability Graphs of Petri Nets

Abstract

The question is considered - which graphs are isomorphic to the reachability graphs of Petri nets. Reachability graphs, or sets of achievable states, represent sets of all possible different network states resulting from a given initial state s₀ by a finite chain of permissible transitions. They have a natural structure of an oriented graph with a dedicated initial state, all other states of which are reachable from the initial one, taking into account orientation. At the same time, if the network transitions are marked, the reachability graphs also receive the corresponding marks of all arcs. At the same time, the concept of isomorphism of marked graphs arises, but this publication deals only with issues for networks without markings. Even for this simpler case, some questions remain open. The paper proves that any finite directed graph is modeled by a suitable Petri net, that is, it is isomorphic to the reachability graph of the network. For infinite graphs, examples of non-modeled graphs are given.