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Coverings of Bipartite Graphs

  • A. L. Dulmage (a1) and N. S. Mendelsohn (a1)
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For the purpose of analysing bipartite graphs (hereinafter called simply graphs) the concept of an exterior covering is introduced. In terms of this concept it is possible in a natural way to decompose any graph into two parts, an inadmissible part and a core. It is also possible to decompose the core into irreducible parts and thus obtain a canonical reduction of the graph. The concept of irreducibility is very easily and naturally expressed in terms of exterior coverings. The role of the inadmissible edges of a graph is to obstruct certain natural coverings of the graph.

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References
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1. Birkhoff, Garrett, Lattice theory (revised edition) Amer. Math. Soc. Coll. Pub., 25 (1948).
2. Dulmage, A. L. and Mendelsohn, N. S. , Some generalizations of the problem of distinct representatives, Can. J. Math., 10 (1958), 230-241.
3. Dulmage, A. L., The convex hull of sub permutation matrices, Proc. Amer. Math. Soc, 9 (1958), 253-254.
4.Dulmage, A. L. and Halperin, I., On a theorem of Frobenius-König and J. von Neumann's game of hide and seek, Trans. Roy. Soc. Can. Ser. IIi , 49 (1955), 23-29.
5. König, D., Theorie der endlichen und unendlichen Graphen, (Chelsea, New York, 1950).
6. Mann, H. B. and Ryser, H. J., Systems of distinct representatives, Amer. Math. Monthly, 60 (1953), 397-401.
7. Ore, O., Graphs and matching theorems, Duke Math. J., 22 (1955), 625'639.
8. Ryser, H. J., Matrices of zeros and ones, Can. J. Math., 9 (1957), 371-377.
9. von Neumann, J., A certain zero sum two person game equivalent to the optimal assignment problem, Contribution to the theory of games II, Annals of Mathematics Studies, 28, (Princeton, 1953), pp. 5-12.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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