The Max-Flow Min-Cut theorem for countable networks

Ron Aharoni, Eli Berger, Agelos Georgakopoulos, Amitai Perlstein, Philipp Sprüssel

Research output: Contribution to journalArticlepeer-review

Abstract

We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are "orthogonal" to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff's first law also for ends.

Original languageEnglish
Pages (from-to)1-17
Number of pages17
JournalJournal of Combinatorial Theory. Series B
Volume101
Issue number1
DOIs
StatePublished - Jan 2011

Bibliographical note

Funding Information:
E-mail addresses: ra@tx.technion.ac.il (R. Aharoni), berger@cri.haifa.ac.il (E. Berger), georgakopoulos@math.uni-hamburg (A. Georgakopoulos), perlstein@tx.technion.ac.il (A. Perlstein), spruessel@math.uni-hamburg (P. Sprüssel). 1 The research of the first author was supported by grants from the Israel Science Foundation, BSF, the M. & M.L Bank Mathematics Research Fund and the fund for the promotion of research at the Technion and a Seniel Ostrow Research Fund. 2 The research of the second author was supported by a BSF grant. 3 The research of the first, third and fifth authors was supported by a GIF grant.

Keywords

  • Ends of graphs
  • Flows
  • Infinite graphs
  • Max-Flow Min-Cut
  • Networks

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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