Contentment in graph theory: Covering graphs with cliques
James B. Orlin
- Vol. 80, Iss: 5, pp 406-424
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In this article, the minimum number of complete subgraphs of a graph G which include all of the edges of G, and the minimum bipartite subgraph of G which cover G are both shown to be NP-complete.Abstract:
Fundamental questions posed by Boole in 1868 on the theory of sets have in recent years been translated to problems in graph theory. The major problems that this paper deals with are determining the minimum number of complete subgraphs of graph G which include all of the edges of G, and determining the minimum number of complete bipartite subgraphs which cover G. The two problems are of a very similar nature. Determining whether there is a projective plane of order p is a special case of the former problem. The latter problem has a natural translation into matrix theory which yields tight upper and lower bounds. An elementary proof is given for Graham's theorem. Two non-obvious classes are given for which the above problems are easily handled; however, this author doubts that these classes can be extended significantly. Two new problems are shown in this paper to be NP-complete. Finally, several conjectures and unsolved problems are posed within the body of the paper.read more
Citations
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References
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TL;DR: The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible.
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The complexity of theorem-proving procedures
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Methods of operations research
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