2019/04/12 by Attila Joó, Joó, Attila
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1904.06244
openalex publication_date 2019/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Erdős-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 Erdős-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.