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Boxicity and Separation Dimension

  • Manu Basavaraju
  • L. Sunil Chandran
  • Martin Charles GolumbicEmail author
  • Rogers Mathew
  • Deepak Rajendraprasad
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

A family \(\mathcal {F}\) of permutations of the vertices of a hypergraph \(H\) is called pairwise suitable for \(H\) if, for every pair of disjoint edges in \(H\), there exists a permutation in \(\mathcal {F}\) in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for \(H\) is called the separation dimension of \(H\) and is denoted by \(\pi (H)\). Equivalently, \(\pi (H)\) is the smallest natural number \(k\) so that the vertices of \(H\) can be embedded in \(\mathbb {R}^k\) such that any two disjoint edges of \(H\) can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph \(H\) is equal to the boxicity of the line graph of \(H\). This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.

Keywords

Separation dimension Boxicity Scrambling permutation Line graph Acyclic chromatic number 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Manu Basavaraju
    • 1
  • L. Sunil Chandran
    • 2
  • Martin Charles Golumbic
    • 3
    Email author
  • Rogers Mathew
    • 3
  • Deepak Rajendraprasad
    • 3
  1. 1.University of BergenBergenNorway
  2. 2.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia
  3. 3.Department of Computer Science, Caesarea Rothschild InstituteUniversity of HaifaHaifaIsrael

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