Advertisement

Small k-Dominating Sets in Planar Graphs with Applications

  • Cyril Gavoille
  • David Peleg
  • André Raspaud
  • Eric Sopena
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2204)

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 kD/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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hans Leo Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209:1–45, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Victor Chepoi and Yann Vaxes. On covering bridged plane triangulations with balls. Manuscript submitted, 2000.Google Scholar
  3. 3.
    Reinhard Diestel. Graph Theory (second edition), volume 173 of Graduate Texts in Mathematics. Springer, February 2000.Google Scholar
  4. 4.
    David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27:275–291, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cyril Gavoille. On the dilation of interval routing. The Computer Journal, 43(1):1–7, 2000.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cyril Gavoille. A survey on interval routing. Theoretical Computer Science, 245(2):217–253, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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. 8.
    Cyril Gavoille and David Peleg. The compactness of interval routing. SIAM Journal on Discrete Mathematics, 12(4):459–473, October 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 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. 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.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Laszlo Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383–390, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    G. MacGillivray and K. Seyffarth. Domination numbers of planar graphs. Journal of Graph Theory, 22(3):213–229, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Neil Robertson and Paul D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7:309–322, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nicola Santoro and Ramez Khatib. Labelling and implicit routing in networks. The Computer Journal, 28(1):5–8, February 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 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. 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. 17.
    Jan van Leeuwen and Richard B. Tan. Interval routing. The Computer Journal, 30(4):298–307, 1987.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Cyril Gavoille
    • 1
  • David Peleg
    • 2
  • André Raspaud
    • 1
  • Eric Sopena
    • 1
  1. 1.LaBRI, Université Bordeaux ITalence CedexFrance
  2. 2.Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations