On optimal graphs embedded into paths and rings, with analysis using l ₁-spheres

Abstract

In this paper we study path layouts in communication networks. Stated in graph-theoretic terms, these layouts are translated into embeddings (or linear arrangements) of the vertices of a graph with N nodes onto the points 1, 2, ... N of the x-axis. We look for a graph with minimum diameter D ^L _c (N), for which such an embedding is possible, given a bound c on the cutwidth of the embedding. We develop a technique to embed the nodes of such graphs into the integral lattice points in the c-dimensional l ₁-sphere. Using this technique, we show that the minimum diameter D ^L _c (N) satisfies R _c (N) ≤ D ^L _c (N) ≤ 2R _c (N), where R _c (N) is the minimum radius of a c-dimensional l ₁-sphere that contains N points. Extensions of the results to augmented paths and ring networks are also presented. Using geometric arguments, we derive analytical bounds for R _c (N), which result in substantial improvements on some known lower and upper bounds.