Proportional contact representations of 4-connected planar graphs

Md Jawaherul Alam, Stephen G Kobourov

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

1 Citation (Scopus)

Abstract

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 publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages211-223
Number of pages13
Volume7704 LNCS
DOIs
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)03029743
ISSN (Electronic)16113349

Other

Other20th International Symposium on Graph Drawing, GD 2012
CountryUnited States
CityRedmond, WA
Period9/19/129/21/12

Fingerprint

Hamiltonians
Planar graph
Connected graph
Directly proportional
Contact
Polygon
Disprove
Graph in graph theory
Vertex of a graph
Rectangle
Upper and Lower Bounds
Disjoint
Interior
Adjacent
NP-complete problem
Imply
Cycle

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Alam, M. J., & Kobourov, S. G. (2013). Proportional contact representations of 4-connected planar graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7704 LNCS, pp. 211-223). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7704 LNCS). https://doi.org/10.1007/978-3-642-36763-2_19

Proportional contact representations of 4-connected planar graphs. / Alam, Md Jawaherul; Kobourov, Stephen G.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7704 LNCS 2013. p. 211-223 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7704 LNCS).

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

Alam, MJ & Kobourov, SG 2013, Proportional contact representations of 4-connected planar graphs. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 7704 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7704 LNCS, pp. 211-223, 20th International Symposium on Graph Drawing, GD 2012, Redmond, WA, United States, 9/19/12. https://doi.org/10.1007/978-3-642-36763-2_19
Alam MJ, Kobourov SG. Proportional contact representations of 4-connected planar graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7704 LNCS. 2013. p. 211-223. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-36763-2_19
Alam, Md Jawaherul ; Kobourov, Stephen G. / Proportional contact representations of 4-connected planar graphs. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7704 LNCS 2013. pp. 211-223 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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