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2. The techniques rely on new NP-completeness results for hypergraph colorings.
doi:10.1016/0166-218X(93)90225-D 
Copyright © 1993 Published by Elsevier Science B.V. All rights reserved.
Efficient sets in graphs
Received 10 October 1990;
revised 15 April 1991.
Available online 26 March 2002.
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Abstract
The efficiency of a set S of vertices in an undirected graph G=(V,E) is defined to be 
(S)=|{v| vεV—S and v is adjacent to exactly one vertex in S}|, i.e., the number of vertices in V—S that are dominated by exactly one vertex in S. The efficiency of a graph G=(V,E) equals the maximum efficiency of any subset S of vertices of V. A linear time algorithm is presented for computing the efficiency of an arbitrary tree and an NP-completeness proof is given for the problem of deciding if an arbitrary planar bipartite graph has a set S such that (S)≥k, for some positive integer k.
