Proportional contact representations of 4-connected planar graphs

Md Jawaherul Alam, Stephen G. Kobourov

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations


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.

Original languageEnglish (US)
Title of host publicationGraph Drawing - 20th International Symposium, GD 2012, Revised Selected Papers
Number of pages13
StatePublished - 2013
Event20th International Symposium on Graph Drawing, GD 2012 - Redmond, WA, United States
Duration: Sep 19 2012Sep 21 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7704 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other20th International Symposium on Graph Drawing, GD 2012
Country/TerritoryUnited States
CityRedmond, WA

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


Dive into the research topics of 'Proportional contact representations of 4-connected planar graphs'. Together they form a unique fingerprint.

Cite this