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Coverings of Bipartite Graphs

Published online by Cambridge University Press:  20 November 2018

A. L. Dulmage  and
N. S. Mendelsohn
A. L. Dulmage
Affiliation:
University of Manitoba
N. S. Mendelsohn
Affiliation:
University of Manitoba
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 517 - 534
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Birkhoff, Garrett, Lattice theory (revised edition) Amer. Math. Soc. Coll. Pub., 25 (1948).Google Scholar
2. Dulmage, A. L. and Mendelsohn, N. S. , Some generalizations of the problem of distinct representatives, Can. J. Math., 10 (1958), 230-241.Google Scholar
3. Dulmage, A. L., The convex hull of sub permutation matrices, Proc. Amer. Math. Soc, 9 (1958), 253-254.Google Scholar
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.Google Scholar
5. König, D., Theorie der endlichen und unendlichen Graphen, (Chelsea, New York, 1950).Google Scholar
6. Mann, H. B. and Ryser, H. J., Systems of distinct representatives, Amer. Math. Monthly, 60 (1953), 397-401.Google Scholar
7. Ore, O., Graphs and matching theorems,Duke Math. J., 22 (1955), 625'639.Google Scholar
8. Ryser, H. J., Matrices of zeros and ones, Can. J. Math., 9 (1957), 371-377.Google Scholar
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.Google Scholar