Structure of the Reachability Problem for (0, l)-Capacitated Marked Graphs

Many classical results in graph theory are equivalent to the maximum-flow minimum-cut theorem of network-flow theory [l]. These results include Tutte’s characterization of maximum matchings in general graphs, Hall’s theorem on bipartite matching, and Menger’s theorem on connectivity. Equivalence among these problems is established by constructing appropriate (O,l)communication networks which permit flows of values of only zero or one on each of its edges. This equivalence has made possible the design of efficient algorithms for matching and for connectivity analysis because efficient algorithms are available for computing maximum flows in (0, 1)-communication networks. These pioneering works have provided the motivation for the study presented in this paper. We study the structure of the reachability problem for (O,l)capacitated marked graphs and derived a purely graph-theoretic characterization of this problem. We draw attention to related works presented in [2] and [3], which were motivated by applications in two unrelated areas: network synthesis and routing in communications networks. We now present the necessary background material on marked graphs. Though our presentation in this section is in terms of capacitated marked graphs, all of the definitions and results (Theorems 1 and 2) here are easy generalizations of those given in [4] and [5] for uncapacitated graphs. For terms not explicitly defined here, [5] may be consulted. A capacitated marked graph is a marked graph G = (V, E) in which a lower bound L(e) and an upper bound U(e) are specified on the token count M(e) of edge e E E, for all markings of G. A marking M is called feasible if and only if L(e) I M(e) I U(e), Ve E E. The enabling number of a vertex u E V under a marking M of a capacitated marked graph G is defined as