Obtaining a marked graph from a synchronic distance matrix

The notion of synchronic distances is a fundamental concept introduced by Petri [1]. It is a metric closely related to a degree of mutual dependence between two events in a condition/event system. The synchronic distance between two vertices in a marked graph is defined here as the maximum number of times one vertex can fire without firing the other. Marked graphs form a subclass of Petri nets. The synchronic distance matrix of a marked graph M with n vertices is an n × n symmetric matrix D* = [dM*(vi, vj)], where dM*(vi, vj) is the synchronic distance between vertices vi and vj. Conversely, using matrix D, the focus is on finding a marked graph whose synchronic distance matrix is D. Murata et al. [2] show that the synchronic distance matrix of a marked graph is a distance matrix of an undirected network and give a method for finding a marked graph when D is realizable as the distance matrix of a tree with positive integer edge weights. This paper presents a necessary and sufficient condition for D to be the synchronic distance matrix of a live marked graph. A matrix D is given such that D is a distance matrix of an undirected network and there does not exist a marked graph whose synchronic distance matrix is D. Also, a method is proposed for finding a live marked graph when D is realizable as the distance matrix of a tree as well as that of a nontree graph like a cycle.