Proportional Contact Representations of 4-Connected Planar Graphs

  • Md. Jawaherul Alam
  • Stephen G. Kobourov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


In a contact representation of a planar graph, vertices are represented by interior-disjoint polygons and two polygons share a non-empty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation is called proportional if each polygon realizes an area proportional to the vertex weight. In this paper we study proportional contact representations of 4-connected internally triangulated planar graphs. The best known lower and upper bounds on the polygonal complexity for such graphs are 4 and 8, respectively. We narrow the gap between them by proving the existence of a representation with complexity 6. We then disprove a 10-year old conjecture on the existence of a Hamiltonian canonical cycle in a 4-connected maximal planar graph, which also implies that a previously suggested method for constructing proportional contact representations of complexity 6 for these graphs will not work. Finally we prove that it is NP-hard to decide whether a 4-connected planar graph admits a proportional contact representation using only rectangles.


Planar Graph Hamiltonian Cycle Pivot Point Contact Representation Maximal Segment 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • Stephen G. Kobourov
    • 1
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA

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