# Centre for Discrete and Applicable Mathematics

## CDAM Research Report, LSE-CDAM-2004-15

August 2004

The External Network Problem

Jan van den Heuvel and Matthew Johnson

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  p1 = v1...vsp2 = w1...wt  in D such that vs and w1 are external vertices (the idea is that v1 can contact wt along this path using an external link from vs to w1). Then D 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 must be externally-k-arc-strong.

We also consider two generalisations of the problem: first we suppose that the number of arc-disjoint paths required from u to v may differ for each choice of u and v, and secondly we suppose that each vertex has a cost and that we are required to find a minimum cost set of external vertices. We show that these two problems are NP-hard.

Finally, we consider the analogue of this problem for vertex-connectivity in undirected graphs. A graph G with set of external vertices X is externally-k-connected if there are k vertex-disjoint paths joining each pair of vertices in G. We present polynomial algorithms for finding a minimum size set of external vertices subject to the requirement that G must be externally-k-connected for the cases  k in {1,2,3}.

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