Skip to main content
SpringerLink
Log in

Small k-Dominating Sets in Planar Graphs with Applications

  • Conference paper
  • First Online:
Book cover Graph-Theoretic Concepts in Computer Science (WG 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2204))

Included in the following conference series:

Abstract

A subset of nodes S in a graph G is called k-dominating if, for every node u of the graph, the distance from u to S is at most k. We consider the parameter γk(G) defined as the cardinality of the smallest k-dominating set of G. For planar graphs, we show that for every ε > 0 and for every k ≽ (5/7 + ε)D, γk(G) = O(1/ε). For several subclasses of planar graphs of diameter D, we show that γk(G) is bounded by a constant for k ≽ D/2. We conjecture that the same result holds for every planar graph. This problem is motivated by the design of routing schemes with compact data structures.

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

Access this chapter

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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hans Leo Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209:1–45, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  2. Victor Chepoi and Yann Vaxes. On covering bridged plane triangulations with balls. Manuscript submitted, 2000.

    Google Scholar 

  3. Reinhard Diestel. Graph Theory (second edition), volume 173 of Graduate Texts in Mathematics. Springer, February 2000.

    Google Scholar 

  4. David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27:275–291, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  5. Cyril Gavoille. On the dilation of interval routing. The Computer Journal, 43(1):1–7, 2000.

    Article  MathSciNet  Google Scholar 

  6. Cyril Gavoille. A survey on interval routing. Theoretical Computer Science, 245(2):217–253, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  7. Cyril Gavoille, Michal Katz, Nir A. Katz, Christophe Paul, and David Peleg. Approximate distance labeling schemes. In 9th Annual European Symposium on Algorithms (ESA), volume Lectures Notes in Computer Science. Springer, August 2001. To appear.

    Google Scholar 

  8. Cyril Gavoille and David Peleg. The compactness of interval routing. SIAM Journal on Discrete Mathematics, 12(4):459–473, October 1999.

    Article  MATH  MathSciNet  Google Scholar 

  9. Cyril Gavoille, David Peleg, André Raspaud, and Eric Sopena. Small k-dominating sets in planar graphs with applications. Research Report RR-1258-01, LaBRI, University of Bordeaux, 351, cours de la Libération, 33405 Talence Cedex, France, May 2001.

    Google Scholar 

  10. Rastislav Kráľovič, Peter Ružička, and Daniel Štefankovič. The complexity of shortest path and dilation bounded interval routing. Theoretical Computer Science, 234(1–2):85–107, 2000.

    MATH  MathSciNet  Google Scholar 

  11. Laszlo Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383–390, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. MacGillivray and K. Seyffarth. Domination numbers of planar graphs. Journal of Graph Theory, 22(3):213–229, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  13. Neil Robertson and Paul D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7:309–322, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  14. Nicola Santoro and Ramez Khatib. Labelling and implicit routing in networks. The Computer Journal, 28(1):5–8, February 1985.

    Article  MATH  MathSciNet  Google Scholar 

  15. Savio S. H. Tse and Francis C. M. Lau. An optimal lower bound for interval routing in general networks. In 4th International Colloquium on Structural Information & Communication Complexity (SIROCCO), pages 112–124. Carleton Scientific, July 1997.

    Google Scholar 

  16. Savio S. H. Tse and Francis C. M. Lau. Some results on the space requirement of interval routing. In 6th International Colloquium on Structural Information & Communication Complexity (SIROCCO), pages 264–279. Carleton Scientific, July 1999.

    Google Scholar 

  17. Jan van Leeuwen and Richard B. Tan. Interval routing. The Computer Journal, 30(4):298–307, 1987.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LaBRI, Université Bordeaux I, 351, cours de la Libération, 33405, Talence Cedex, France

    Cyril Gavoille, André Raspaud & Eric Sopena

  2. Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    David Peleg

Authors
  1. Cyril Gavoille
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Peleg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. André Raspaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eric Sopena
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Rostock, 18051, Rostock, Germany

    Andreas Brandstädt  & Van Bang Le  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gavoille, C., Peleg, D., Raspaud, A., Sopena, E. (2001). Small k-Dominating Sets in Planar Graphs with Applications. In: Brandstädt, A., Le, V.B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2001. Lecture Notes in Computer Science, vol 2204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45477-2_19

Download citation

  • DOI: https://doi.org/10.1007/3-540-45477-2_19

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42707-0

  • Online ISBN: 978-3-540-45477-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation