Polynomial-Time Analysis of Toroidal Periodic Graphs

Abstract

A toroidal periodic graph GD 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 v at position z in the periodic graph if and only if there is an edge from u to v with vector t in the static graph such that the difference z−(y+t) is the sum of integral multiples of columns of D. We show that (1) the general path problem in toroidal periodic graphs can be solved with methods from linear integer programming, (2) path problems for toroidal periodic graphs GD can be solved in polynomial time if G has a bounded number of strongly connected components, (3) the number of strongly connected components in a toroidal periodic graph can be determined in polynomial time, and (4) a periodic description for each strongly connected component of GD can be found in polynomial time.