NP-completeness for minimizing maximum edge length in grid embeddings

Abstract

Given an embedding f: G → Z 2 of a graph G in the two-dimensional lattice, let | f| be the maximum L 1 distance between points f( x) and f( y) where xy is an edge of G. Let B 2( G) be the minimum | f| over all embeddings f. It is shown that the determination of B 2( G) for arbitrary G is NP-complete. Essentially the same proof can be used in showing the NP-completeness of minimizing | f| over all embeddings f: G → Z n of G into the n-dimensional integer lattice for any fixed n ≥ 2.