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arXiv · 1904.06244

The Complete Lattice of Erd\H{o}s-Menger Separations

Attila Joó
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Abstract

F. Escalante and T. Gallai studied in the seventies the structure of different kind of separations and cuts between a vertex pair in a (possibly infinite) graph. One of their results is that if there is a finite separation, then the optimal (i.e. minimal sized) separations form a finite distributive lattice with respect to a natural partial order. Furthermore, any finite distributive lattice can be represented this way. If there is no finite separation then cardinality is a too rough measure to capture being 'optimal'. Menger's theorem provides a structural characterization of optimality if there is a finite separation. We use this characterization to define Erd\H{o}s-Menger separations even if there is no finite separation. The generalization of Menger's theorem to infinite graphs (which was not available until 2009) ensures that Erd\H{o}s-Menger separations always exist. We show that they form a complete lattice with respect to the partial order given by Escalante and every complete lattice can be represented this way.

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Attila Joó. 2019-04-12. The Complete Lattice of Erd\H{o}s-Menger Separations. https://arxiv.org/abs/1904.06244

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