Skip to main content
SpringerLink
Log in
Book cover

Scandinavian Workshop on Algorithm Theory

SWAT 2002: Algorithm Theory — SWAT 2002 pp 140–149Cite as

Powers of Geometric Intersection Graphs and Dispersion Algorithms

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2368))

Abstract

We study powers of certain geometric intersection graphs: interval graphs, m-trapezoid graphs and circular-arc graphs. We define the pseudo product, (G,G′) → G * G′, of two graphs G and G′ on the same set of vertices, and show that G*G′ is contained in one of the three classes of graphs mentioned here above, if both G and G′ are also in that class and fulfill certain conditions. This gives a new proof of the fact that these classes are closed under taking power; more importantly, we get efficient methods for computing the representation for G k if k ≥ 1 is an integer and G belongs to one of these classes, with a given representation sorted by endpoints. We then use these results to give efficient algorithms for the k-independent set, dispersion and weighted dispersion problem on these classes of graphs, provided that their geometric representations are given.

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. G. Agnarsson. On Powers of some Intersection Graphs, Congressus Numerantium, submitted in April 2001.

    Google Scholar 

  2. G. Agnarsson, R. Greenlaw, M. M. Halldórsson. On Powers of Chordal Graphs and Their Colorings, Congressus Numerantium, 144:41–65, (2000).

    MATH  MathSciNet  Google Scholar 

  3. B. K. Bhattacharya, M. E. Houle. Generalized Maximum Independent Sets for Trees. Computing-The 3rd Australian Theory Symposium CATS’97.

    Google Scholar 

  4. B. K. Bhattacharya, M. E. Houle. Generalized Maximum Independent Sets for Trees in Subquadratic Time. 10th Int. Symp. on Algorithms and Computation ISAAC’99, LNCS 1741 (Springer), 435–445.

    Google Scholar 

  5. E. Dahlhaus and P. Duchet. On Strongly Chordal Graphs. Ars Combinatoria, 24 B:23–30, (1987).

    MathSciNet  Google Scholar 

  6. P. Damaschke. Distances in Cocomparability Graphs and Their Powers. Discrete Applied Mathematics, 35:67–72, (1992).

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Damaschke. Efficient Dispersion Algorithms for Geometric Intersection Graphs. 26th Int. Workshop on Graph-Theoretic Concepts in Computer Science WG’2000, LNCS 1928 (Springer), 107–115.

    Chapter  Google Scholar 

  8. S. Felsner, R. Müller and L. Wernisch. Trapezoid Graphs and Generalizations, Geometry and Algorithms. Discrete Applied Mathematics, 74:13–32, (1997).

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Flotow. On Powers of m-Trapezoid Graphs. Discrete Applied Mathematics, 63:187–192, (1995).

    Article  MATH  MathSciNet  Google Scholar 

  10. C. Flotow. On Powers of Circular Arc graphs and Proper Circular Arc Graphs. Discrete Applied Mathematics, 74:199–207, (1996).

    Article  MathSciNet  Google Scholar 

  11. J. Håstad. Clique is hard to approximate within n1-∈. Acta Mathematica, 182:105–142, (1999).

    Article  MATH  MathSciNet  Google Scholar 

  12. W. L. Hsu, K. H. Tsai. Linear-time Algorithms on Circular Arc Graphs. Information Proc. Letters, 40:123–129, (1991).

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Y. Hsiao, C. Y. Tang, R. S. Chang. An Efficient Algorithm for Finding a Maximum Weight 2-Independent Setw on Interval Graphs. Information Proc. Letters, 43:229–235, (1992).

    Article  MATH  MathSciNet  Google Scholar 

  14. D. T. Lee, M. Sarrafzadeh, Y. F. Wu. Minimum Cuts for Circular-Arc Graphs. SIAM J. Computing, 19:1041–1050, (1990).

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Masuda and K. Nakajima. An Optimal Algorithm for Finding a Maximum Independent Set of a Circular-Arc Graph. SIAM J. Computing, 17:219–230, (1988).

    Article  MathSciNet  Google Scholar 

  16. A. Raychaudhuri. On Powers of Interval and Unit Interval Graphs. Congressus Numerantium, 59:235–242, (1987).

    MathSciNet  Google Scholar 

  17. A. Raychaudhuri. On Powers of Strongly Chordal and Circular Graphs. Ars Combinatoria, 34:147–160, (1992).

    MATH  MathSciNet  Google Scholar 

  18. D. J. Rosenkrantz, G. K. Tayi, S. S. Ravi. Facility Dispersion Problems Under Capacity and Cost Constraints. J. of Combinatorial Optimization, 4:7–33 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  19. D. B. West. Introduction to Graph Theory. Prentice-Hall Inc., Upper Saddle River, New Jersey, (1996).

    MATH  Google Scholar 

  20. M. Yannakakis. The complexity of the partial order dimension problem. SIAM J. Alg. Disc. Meth. 3:351–358, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Q. Zheng. Maximum Independent Sets of Circular Arc Graphs: Simplified Algorithms and Proofs. Networks, 28:15–19, (1996).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Armstrong Atlantic State University, Savannah, Georgia, 31419-1997

    Geir Agnarsson

  2. Department of Computer Science, Chalmers University, 41296, Göteborg, Sweden

    Peter Damaschke

  3. Department of Computer Science, University of Iceland, IS-107, Reykjavík, Iceland

    Magnús M. Halldórsson

  4. Iceland Genomics Corp. (UVS), Snorrabraut 60, IS-105, Reykjavik

    Magnús M. Halldórsson

Authors
  1. Geir Agnarsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Damaschke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Magnús M. Halldórsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Applied Mathematics, University of Kuopio, P.O. Box 1627, 70211, Kuopio, Finland

    Martti Penttonen

  2. BRICS, University of Aarhus, Department of Computer Science, NY Munkegade, 8000, Aarhus C, Denmark

    Erik Meineche Schmidt

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Agnarsson, G., Damaschke, P., Halldórsson, M.M. (2002). Powers of Geometric Intersection Graphs and Dispersion Algorithms. In: Penttonen, M., Schmidt, E.M. (eds) Algorithm Theory — SWAT 2002. SWAT 2002. Lecture Notes in Computer Science, vol 2368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45471-3_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-45471-3_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43866-3

  • Online ISBN: 978-3-540-45471-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation