Document Open Access Logo

Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs

Authors Giordano Da Lozzo, William E. Devanny, David Eppstein, Timothy Johnson

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2017.24.pdf
  • Filesize: 0.92 MB
  • 14 pages

Document Identifiers

Subject Classification

Keywords
  • Square-Contact Representations
  • Partial 2-Trees
  • Simply-Nested Graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A square-contact representation of a planar graph G = (V,E) maps vertices in V to interior-disjoint axis-aligned squares in the plane and edges in E to adjacencies between the sides of the corresponding squares. In this paper, we study proper square-contact representations of planar graphs, in which any two squares are either disjoint or share infinitely many points.

We characterize the partial 2-trees and the triconnected cycle-trees allowing for such representations. For partial 2-trees our characterization uses a simple forbidden subgraph whose structure forces a separating triangle in any embedding. For the triconnected cycle-trees, a subclass of the triconnected simply-nested graphs, we use a new structural decomposition for the graphs in this family, which may be of independent interest. Finally, we study square-contact representations of general triconnected simply-nested graphs with respect to their outerplanarity index.

Cite As Get BibTex

Giordano Da Lozzo, William E. Devanny, David Eppstein, and Timothy Johnson. Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ISAAC.2017.24

Author Details

Giordano Da Lozzo
William E. Devanny
David Eppstein
Timothy Johnson

References

  1. Md. Jawaherul Alam, David Eppstein, Michael Kaufmann, Stephen G. Kobourov, Sergey Pupyrev, André Schulz, and Torsten Ueckerdt. Contact graphs of circular arcs. In WADS '15, volume 9214 of LNCS, pages 1-13. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21840-3_1.
  2. Clinton Bowen, Stephane Durocher, Maarten Löffler, Anika Rounds, André Schulz, and Csaba D. Tóth. Realization of simply connected polygonal linkages and recognition of unit disk contact trees. In GD '15, volume 9411 of LNCS, pages 447-459. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-27261-0_37.
  3. Steven Chaplick, Stephen G. Kobourov, and Torsten Ueckerdt. Equilateral l-contact graphs. In WG '13, volume 8165 of LNCS, pages 139-151. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-45043-3_13.
  4. Robert J. Cimikowski. Finding hamiltonian cycles in certain planar graphs. Inf. Process. Lett., 35(5):249-254, 1990. URL: http://dx.doi.org/10.1016/0020-0190(90)90053-Z.
  5. Giordano Da Lozzo, William Devanny, David Eppstein, and Timothy Johnson. Square-contact representations of partial 2-trees and triconnected simply-nested graphs. Tech. Report arXiv:1710.00426, Cornell University, 2017. URL: http://arxiv.org/abs/1710.00426.
  6. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999. Google Scholar
  7. Stefan Felsner and Mathew C. Francis. Contact representations of planar graphs with cubes. In SoCG '11, pages 315-320. ACM, 2011. URL: http://dx.doi.org/10.1145/1998196.1998250.
  8. Daniel Gonçalves, Benjamin Lévêque, and Alexandre Pinlou. Triangle contact representations and duality. Discrete Comput. Geom., 48(1):239-254, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9400-1.
  9. H. Harborth. Lösung zu Problem 664A. Elemente der Mathematik, 29:14-15, 1974. Google Scholar
  10. Petr Hliněný. Contact graphs of line segments are NP-complete. Discrete Math., 235(1-3):95-106, 2001. URL: http://dx.doi.org/10.1016/S0012-365X(00)00263-6.
  11. Oded Schramm. Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps. PhD thesis, Princeton University, 1990. Google Scholar
  12. Oded Schramm. Square tilings with prescribed combinatorics. Israel J. Math., 84(1-2):97-118, 1993. URL: http://dx.doi.org/10.1007/BF02761693.
  13. Kenneth Stephenson. Introduction to Circle Packing: The theory of discrete analytic functions. Cambridge University Press (1), 2005. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact