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Interval Graph Representation with Given Interval and Intersection Lengths

  • Johannes Köbler
  • Sebastian Kuhnert
  • Osamu Watanabe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)

Abstract

We consider the problem of finding interval representations of graphs that additionally respect given interval lengths and/or pairwise intersection lengths, which are represented as weight functions on the vertices and edges, respectively. Pe’er and Shamir proved that the problem is \(\text{\upshape\textsf{NP}}\)-complete if only the former are given [SIAM J. Discr. Math. 10.4, 1997]. We give both a linear-time and a logspace algorithm for the case when both are given, and both an \(\ensuremath{\mathcal{O}}(n\cdot m)\) time and a logspace algorithm when only the latter are given. We also show that the resulting interval systems are unique up to isomorphism.

Complementing their hardness result, Pe’er and Shamir give a polynomial-time algorithm for the case that the input graph has a unique interval ordering of its maxcliques. For such graphs, their algorithm computes an interval representation that respects a given set of distance inequalities between the interval endpoints (if it exists). We observe that deciding if such a representation exists is \(\text{\upshape\textsf{NL}}\)-complete.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Johannes Köbler
    • 1
  • Sebastian Kuhnert
    • 1
  • Osamu Watanabe
    • 2
  1. 1.Inst. für InformatikHumboldt-Universität zu BerlinGermany
  2. 2.Dept. of Mathematical and Computing SciencesTokyo Institute of TechnologyJapan

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