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Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

  • Steven ChaplickEmail author
  • Paul Dorbec
  • Jan Kratochvíl
  • Mickael Montassier
  • Juraj Stacho
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.

On the positive side we give two polynomial time algorithms for the grid contact case. The first one is for the case when all vertical segments are pre-represented (note: the problem remains NP-complete when a subset of the vertical segments is specified, even if none of the horizontals are). Secondly, we show that the case when the vertical segments have only their \(x\)-coordinates specified (i.e., they are ordered horizontally) is polynomially equivalent to level planarity, which is known to be solvable in polynomial time.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Steven Chaplick
    • 1
    Email author
  • Paul Dorbec
    • 2
  • Jan Kratochvíl
    • 3
  • Mickael Montassier
    • 4
  • Juraj Stacho
    • 5
  1. 1.Institut Für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.University of Bordeaux, CNRS - LaBRI, UMR 5800TalenceFrance
  3. 3.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic
  4. 4.Université Montpellier 2, CNRS - LIRMMMontpellierFrance
  5. 5.Department of Industrial Engineering and Operations ResearchColumbia UniversityNew YorkUSA

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