Coverings of Bipartite Graphs

@article{Dulmage1958CoveringsOB,
  title={Coverings of Bipartite Graphs},
  author={A. L. Dulmage and N. S. Mendelsohn},
  journal={Canadian Journal of Mathematics},
  year={1958},
  volume={10},
  pages={517 - 534}
}
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… 
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References

SHOWING 1-8 OF 8 REFERENCES

The convex hull of sub-permutation matrices

1. Introduction. A combinatorial theorem [1; 3] usually referred to as "the marriage problem" or "the problem of distinct representa-tives" has the following matrix formulation; the convex hull of…

Some Generalizations of the Problem of Distinct Representatives

If S 1, S 2, S 3, … , S n are subsets of a set M then it is known that a necessary and sufficient condition that it is possible to choose representatives at such that ai is in S i for (i = 1, 2, 3, ……

Matrices of zeros and ones

Let A be a. matrix of m rows and n columns and let the entries of A be the integers 0 and 1. We call such a matrix a (0, 1) -matrix of size m by n. The 2 (0, 1)-matrices of size m by n play a…

Lattice Theory Revised Edition

Theorie der endlichen und unendlichen Graphen

1. A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem

Graphs and matching theorems

On a theorem of Frobenius-Kônig and J. von Neumann's game of hide and seek

  • Trans. Roy. Soc. Can. Ser. I l l ,
  • 1955