Polynomial Time Analysis of Torodial Periodic Graphs

A toroidal periodic graph G D is defined by an integral d = d matrix D and a directed graph G in which the edges are associated with d-dimensional integral vectors. The periodic graph has a vertex for each vertex of the static graph and for each integral position in the parallelpiped defined by the columns of D. There is an edge from vertex u at position y to vertex ̈ at position z in the periodic graph if and only if there is an edge from u to ̈ with vector t in the static graph such Ž . that the difference z y y q t is the sum of integral multiples of columns of D. Ž . We show that 1 the general path problem in toroidal periodic graphs can be Ž . solved with methods from linear integer programming, 2 path problems for toroidal periodic graphs G D can be solved in polynomial time if G has a bounded Ž . number of strongly connected components, 3 the number of strongly connected components in a toroidal periodic graph can be determined in polynomial time, Ž . D and 4 a periodic description for each strongly connected component of G can be found in polynomial time. Q 2000 Academic Press