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A Cut Tree Representation for Pendant Pairs

Authors On-Hei S. Lo, Jens M. Schmidt



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Author Details

On-Hei S. Lo
  • Institut für Mathematik, Technische Universität Ilmenau, Weimarer Strasse 25, D-98693 Ilmenau, Germany
Jens M. Schmidt
  • Institut für Mathematik, Technische Universität Ilmenau, Weimarer Strasse 25, D-98693 Ilmenau, Germany

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On-Hei S. Lo and Jens M. Schmidt. A Cut Tree Representation for Pendant Pairs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 38:1-38:9, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.38

Abstract

Two vertices v and w of a graph G are called a pendant pair if the maximal number of edge-disjoint paths in G between them is precisely min{d(v),d(w)}, where d denotes the degree function. The importance of pendant pairs stems from the fact that they are the key ingredient in one of the simplest and most widely used algorithms for the minimum cut problem today. Mader showed 1974 that every simple graph with minimum degree delta contains Omega(delta^2) pendant pairs; this is the best bound known so far. We improve this result by showing that every simple graph G with minimum degree delta >= 5 or with edge-connectivity lambda >= 4 or with vertex-connectivity kappa >= 3 contains in fact Omega(delta |V|) pendant pairs. We prove that this bound is tight from several perspectives, and that Omega(delta |V|) pendant pairs can be computed efficiently, namely in linear time when a Gomory-Hu tree is given. Our method utilizes a new cut tree representation of graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph algorithms
Keywords
  • Pendant Pairs
  • Pendant Tree
  • Maximal Adjacency Ordering
  • Maximum Cardinality Search
  • Testing Edge-Connectivity
  • Gomory-Hu Tree

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References

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