Skip to main content
SpringerLink
Log in
Book cover

Workshop on Algorithms and Data Structures

WADS 2011: Algorithms and Data Structures pp 583–594Cite as

Planar Subgraphs without Low-Degree Nodes

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6844))

Abstract

We study the following problem: given a geometric graph \(\mathcal{G}\) and an integer k, determine if \(\mathcal{G}\) has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If \(\mathcal{G}\) is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 2) leads to NP-hard problems.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abellanas, M., García, A., Hurtado, F., Tejel, J., Urrutia, J.: Augmenting the connectivity of geometric graphs. Computational Geometry: Theory and Applications 40(3), 220–230 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Jubeh, M., Ishaque, M., Rédei, K., Souvaine, D.L., Tóth, C.D.: Tri-edge-connectivity augmentation for planar straight line graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 902–912. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Barrière, L., Fraigniaud, P., Narayanan, L., Opatrny, J.: Robust position-based routing in wireless ad hoc networks with irregular transmission ranges. Wireless Communications and Mobile Computing 3(2), 141–153 (2003)

    Article  Google Scholar 

  4. Gabriel, K.R., Sokal, R.R.: A new statistical approach to geographic variation analysis. Systematic Zoology 18(3), 259–278 (1969)

    Article  Google Scholar 

  5. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman and Company, New York (1979)

    MATH  Google Scholar 

  6. Kranakis, E., Krizanc, D., Ponce, O.M., Stacho, L.: Bounded length, 2-edge augmentation of geometric planar graphs. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 385–397. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Kranakis, E., Morales Ponce, O., Suomela, J.: Planar subgraphs without low-degree notes (2011), http://www.iki.fi/jukka.suomela/low-degree

  8. Liebers, A.: Planarizing graphs—a survey and annotated bibliography. Journal of Graph Algorithms and Applications 5(1), 1–74 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rutter, I., Wolff, A.: Augmenting the connectivity of planar and geometric graphs. Electronic Notes in Discrete Mathematics 31, 53–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tóth, C.D.: Connectivity augmentation in plane straight line graphs. Electronic Notes in Discrete Mathematics 31, 49–52 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tóth, C.D., Valtr, P.: Augmenting the edge connectivity of planar straight line graphs to three. In: Proc. 13th Spanish Meeting on Computational Geometry, EGC 2009 (2009)

    Google Scholar 

  12. Toussaint, G.T.: The relative neighbourhood graph of a finite planar set. Pattern Recognition 12(4), 261–268 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, Canada

    Evangelos Kranakis & Oscar Morales Ponce

  2. Helsinki Institute for Information Technology HIIT, University of Helsinki, Finland

    Jukka Suomela

Authors
  1. Evangelos Kranakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oscar Morales Ponce
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jukka Suomela
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computer Science, Parallel Computing and Bioinformatics Laboratory, Carleton University, VISM Building, Room 6210, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada

    Frank Dehne

  2. Department of Computer Science and Engineering, Polytechnic Institute of New York University, 5 MetroTech Center, 11201, New York, NY, USA

    John Iacono

  3. School of Computer Science, Herzberg Laboratories, Carleton University, Herzberg Building, Room 5350, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada

    Jörg-Rüdiger Sack

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kranakis, E., Morales Ponce, O., Suomela, J. (2011). Planar Subgraphs without Low-Degree Nodes. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_49

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22300-6_49

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22299-3

  • Online ISBN: 978-3-642-22300-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation