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On the Number of Minimal Separators in Graphs

  • Serge GaspersEmail author
  • Simon Mackenzie
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

We consider the largest number of minimal separators a graph on n vertices can have.

  • We give a new proof that this number is in \(O\left( \left( \frac{1+\sqrt{5}}{2}\right) ^n\cdot n \right) \).

  • We prove that this number is in \(\omega \left( 1.4457^n \right) \), improving on the previous best lower bound of \(\varOmega (3^{n/3}) \subseteq \omega (1.4422^n)\).

This gives also an improved lower bound on the number of potential maximal cliques in a graph. We would like to emphasize that our proofs are short, simple, and elementary.

Keywords

Maximal Clique Horizontal Layer Chordal Graph Infinite Family Minimal Separator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

We thank Yota Otachi and Hitoshi Iwai for pointing out an issue in an earlier version of the lower bound proof.

NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program. Serge Gaspers is the recipient of an Australian Research Council Discovery Early Career Researcher Award (project number DE120101761) and a Future Fellowship (project number FT140100048).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.The University of New South WalesSydneyAustralia
  2. 2.NICTASydneyAustralia

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