Skip to main content

Algorithms on Graphs with Small Dominating Targets

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 4288)

Abstract

A dominating target of a graph G=(V,E) is a set of vertices T s.t. for all W ⊆ V, if T ⊆ W and induced subgraph on W is connected, then W is a dominating set of G. The size of the smallest dominating target is called dominating target number of the graph, dt(G). We provide polynomial time algorithms for minimum connected dominating set, Steiner set, and Steiner connected dominating set in dominating-pair graphs (i.e., dt(G)=2). We also give approximation algorithm for minimum connected dominating set with performance ratio 2 on graphs with small dominating targets. This is a significant improvement on appxd(opt + 2) given by Fomin et.al. [2004] on graphs with small d-octopus.

Classification: Dominating target, d-octopus, Dominating set, Dominating-pair graph, Steiner tree.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
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. Guha, S., Khuller, S.: Approximation Algorithms for Connected Dominating Sets. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 179–193. Springer, Heidelberg (1996)

    Google Scholar 

  2. Balakrishnan, H., Rajaraman, A., Rangan, C.P.: Connected Domination and Steiner Set on Asteroidal Triple-Free Graphs. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 131–141. Springer, Heidelberg (1993)

    Google Scholar 

  3. Robins, G., Zelikovsky, A.: Improved Steiner Tree Approximation in Graphs. In: Proceedings of SODA. LNCS, pp. 770–779. Springer, Heidelberg (2000)

    Google Scholar 

  4. Guha, S., Khuller, S.: Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets. In: Proceedings of FSTTCS year. LNCS, pp. 54–65. Springer, Heidelberg (1998)

    Google Scholar 

  5. Prömel, H.J., Steger, A.: RNC-Approximation Algorithms for the Steiner Problem. In: Proceedings of STACS. LNCS, pp. 559–570. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  6. Clementi, A.E.F., Trevisan, L.: Improved Non-Approximability Results for Vertex Cover with Density Constraints. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 333–342. Springer, Heidelberg (1996)

    Google Scholar 

  7. Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1978)

    Google Scholar 

  8. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (2003)

    Google Scholar 

  9. Kahng, A.B., Robins, G.: On Optimal Interconnections for VLSI. Kluwer Academic, Dordrecht (1995)

    MATH  Google Scholar 

  10. Brandstädt, A., Lee, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)

    Google Scholar 

  11. Caldwell, A., Kahng, A., Mantik, S., Markov, I., Zelikovsky, A.: On Wirelength Estimations for Row-Biased Placement. In: Proceedings of International Symposium on Physical Design, pp. 4–11 (1998)

    Google Scholar 

  12. Kloks, T., Kratsch, D., Müller, H.: On the Structure of Graphs with Bounded Asteroidal Number. Graphs and Combinatorics 17, 295–306 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Korte, B., Prömel, H.J., Steger, A.: Steiner Trees in VLSI Layouts. J. Paths flows and VLSI layout (1990)

    Google Scholar 

  14. Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A Polynomial-Time Approximation Scheme for the Minimum-Connected Dominating Set in Ad Hoc Wireless Networks. Journal of Networks 42(4), 202–208 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal Triple-Free Graphs. SIAM J. Discrete Math. 10(3), 399–430 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Even, S., Pnueli, A., Lempel, A.: Permutation Graphs and Transitive Graphs. J. ACM 19(3), 400–410 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lekkerkerker, C.G., Boland, J.Ch.: Boland: Representation of a Finite Graphs by a Set of Intervals on the Real Line. J. Fund. Math. 51, 245–264 (1962)

    MathSciNet  Google Scholar 

  18. Motwani, R.: Lecture Notes on Approximation Algorithms, Dept. of Comp. Sc., Stanford University, vol. I (1992)

    Google Scholar 

  19. Fomin, F.V., Kratsch, D., Müller, H.: Algorithms for Graphs with Small Octopus. Journal of Discrete Applied Mathematics 134, 105–128 (2004)

    Article  MATH  Google Scholar 

  20. Kratsch, D., Spinrad, J.: Between O(nm) and O(n α). In: Prodeedings of SODA. LNCS, pp. 709–716. Springer, Heidelberg (2003)

    Google Scholar 

  21. Habib, M.: Substitution des Structures Combinatoires, Theorie et Algorithmes. These D’etat, Paris VI (1981)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Aggarwal, D., Dubey, C.K., Mehta, S.K. (2006). Algorithms on Graphs with Small Dominating Targets. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_16

Download citation

  • DOI: https://doi.org/10.1007/11940128_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49694-6

  • Online ISBN: 978-3-540-49696-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics