Edge-Weighted contact representations of planar graphs

  • Authors:

  • Martin Nöllenburg;Roman Prutkin;Ignaz Rutter

  • Affiliations:

  • Institut für Theoretische Informatik, Karlsruhe Institute of Technology, Germany;Institut für Theoretische Informatik, Karlsruhe Institute of Technology, Germany;Institut für Theoretische Informatik, Karlsruhe Institute of Technology, Germany

  • Venue:
  • GD'12 Proceedings of the 20th international conference on Graph Drawing
  • Year:
  • 2012

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Abstract

We study contact representations of edge-weighted planar graphs, where vertices are rectangles or rectilinear polygons and edges are polygon contacts whose lengths represent the edge weights. We show that for any given edge-weighted planar graph whose outer face is a quadrangle, that is internally triangulated and that has no separating triangles we can construct in linear time an edge-proportional rectangular dual if one exists and report failure otherwise. For a given combinatorial structure of the contact representation and edge weights interpreted as lower bounds on the contact lengths, a corresponding contact representation that minimizes the size of the enclosing rectangle can be found in linear time. If the combinatorial structure is not fixed, we prove NP-hardness of deciding whether a contact representation with bounded contact lengths exists. Finally, we give a complete characterization of the rectilinear polygon complexity required for representing biconnected internally triangulated graphs: For outerplanar graphs complexity 8 is sufficient and necessary, and for graphs with two adjacent or multiple non-adjacent internal vertices the complexity is unbounded.