Minimal Obstructions for Partial Representations of Interval Graphs

  • Pavel Klavík
  • Maria Saumell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)


Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation.

In this paper, we characterize the minimal obstructions which make a partial representation non-extendible. This generalizes Lekkerkerker and Boland’s characterization of minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to the first polynomial-time certifying algorithm for partial representation extension of intersection graphs.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Computer Science InstituteCharles University in PraguePragueCzech Republic
  2. 2.Department of Mathematics and European Centre of Excellence NTISUniversity of West BohemiaPilsenCzech Republic

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