Paths and cycles in finite periodic graphs

Abstract

We consider finite periodic graphs Gm defined by nonnegative integer vectors m and directed graphs G whose edges are labeled with integer vector-weights. Gm has a vertex (u, x) for each vertex u of G and each nonnegative integer vector x less than or equal to m. Gm has an edge from (u, x) to (v, x + z) if and only if G has an edge from u to v with vector weight z. We analyze the complexity and present algorithms for finding paths and cycles in finite periodic graphs. The present paper shows that path and cycle problems on finite periodic graphs are PSPACE-complete under various restrictions, but solvable in polynomial time if the vector weights of the edges are bounded.