Polynomial time analysis of toroidal periodic graphs

Abstract

A toroidal periodic graph Gα is defined by a positive integer vector α and a directed graph G in which the edges are associated with integer vectors. Gα has a vertex (v, y) for each vertex v of G and each integer vector (Formula presented) has an edge from (v, y) to (w, z) if and only if G has an edge from v to w associated with t, and z = y+t mod α. We show that path problems for toroidal periodic graphs Gα can be solved in polynomial time if G has a constant number of strongly connected components. The general path problem in toroidal periodic graphs is shown to be NP-complete for all (Formula presented). Additionally, we present a procedure for determining the number of strongly connected components in a toroidal periodic graph. This procedure takes polynomial time for all instances G and α. The introduced methods are very general and can also be used to solve further graph problems in polynomial time on even more general toroidal periodic graphs.