The External Network Problem with Edge- or Arc-Connectivity Requirements

Abstract

The connectivity of a communications network can often be enhanced if the nodes are able, at some expense, to form links using an external network. In this paper, we consider the problem of how to obtain a prescribed level of connectivity with a minimum number of nodes connecting to the external network.

Let D = (V,A) be a digraph. A subset X of vertices in V may be chosen, the so-called external vertices. An internal path is a normal directed path in D; an external path is a pair of internal paths p ₁=v ₁ ⋯ v _s , p ₂=w ₁ ⋯ w _t in D such that v _s and w ₁ are external vertices ( the idea is that v ₁ can contact w _t along this path using an external link from v _t to w ₁ ). Then (D,X) is externally-k-arc-strong if for each pair of vertices u and v in V, there are k arc-disjoint paths ( which may be internal or external ) from u to v.

We present polynomial algorithms that, given a digraph D and positive integer k, will find a set of external vertices X of minimum size subject to the requirement that (D,X) must be externally-k-arc-strong.

Research supported by EPSRC MathFIT grant no. GR/R83514/01.