Extending Partial Representations of Function Graphs and Permutation Graphs

  • Pavel Klavík
  • Jan Kratochvíl
  • Tomasz Krawczyk
  • Bartosz Walczak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)

Abstract

Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear.

We focus on the problem of extending partial representations, which generalizes the recognition problem. We observe that for permutation graphs an easy extension of Golumbic’s comparability graph recognition algorithm can be exploited. This approach fails for function graphs. Nevertheless, we present a polynomial-time algorithm for extending a partial representation of a graph by functions defined on the entire interval [0,1] provided for some of the vertices. On the other hand, we show that if a partial representation consists of functions defined on subintervals of [0,1], then the problem of extending this representation to functions on the entire interval [0,1] becomes NP-complete.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pavel Klavík
    • 1
  • Jan Kratochvíl
    • 1
  • Tomasz Krawczyk
    • 2
  • Bartosz Walczak
    • 2
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityCzech Republic
  2. 2.Theoretical Computer Science Department, Faculty of Mathematics and Computer ScienceJagiellonian UniversityPoland

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