ISAAC 2012: Algorithms and Computation pp 517-526

# Interval Graph Representation with Given Interval and Intersection Lengths

• Johannes Köbler
• Sebastian Kuhnert
• Osamu Watanabe
Conference paper
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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© 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