ICALP 1994: Automata, Languages and Programming pp 544-555

# Polynomial time analysis of toroidal periodic graphs

• F. Höfting
• E. Wanke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 820)

## Abstract

A toroidal periodic graph G α is defined by a positive integer vector α and a directed graph G in which the edges are associated with integer vectors. G α has a vertex (v, y) for each vertex v of G and each integer vector $$\vec 0 \le y < \alpha$$. G α 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 G α 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 $$\alpha \ge \vec 2$$. 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.

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