Suppose you're consulting for a company in northern New Jersey that designs communication networks, and they come to you with the following problem. They re studying a specific n-node communicationnetwork, modeled as a directed graph G = (V, E). For reasons of fault tolerance, they want to divide up G into as many virtual "domains" as possible: A domain in G is a set X of nodes, of size at least 2, so that for each pair of nodes u , v e X there are directed paths from u to v and v to u that are contained entirely in X.
Show that the following Domain Decomposition Problem is NP-com-plete. Given a directed graph G = (V, E) and a number k, can V be partitioned into at least k sets, each of which is a domain?
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