Skip to main content
SpringerLink
Log in

Domatic Partition on Several Classes of Graphs

  • Conference paper
Combinatorial Optimization and Applications (COCOA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7402))

Abstract

The domatic number of a graph G, denoted by DN(G), is the maximum number k such that V can be partitioned into k disjoint dominating sets. The domatic partition problem is to find a partition of the vertices of G into DN(G) dominating sets. The k-domatic partition problem with fixed k is to find a partition of the vertices of G into k dominating sets. In this paper, we show that 3-domatic partition problem is NP-complete on planar bipartite graphs, and the domatic partition problem is NP-complete on co-bipartite graphs. We further show that the unique 3-domatic partition problem is NP-hard on general graphs. Moreover, we propose an O(n)-time algorithm on the 3-domatic partition problem for maximal planar graphs, and O(n 3)-time algorithms on the domatic partition problem for P 4-sparse graphs and tree-cographs, respectively.

This research was supported by National Science Council, Taiwan, under the grant numbers NSC99-2221-E-128-003 and NSC100-2628-E-007-020-MY3.

Every author contributes to this research equally.

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. Bertossi, A.A.: On the domatic number of internal graphs. Information Processing Letters 28, 275–280 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonuccelli, M.A.: Dominating sets and domatic number of circular arc graphs. Discrete Applied Mathematics 12, 203–213 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, G.J.: The domatic number problem. Discrete Mathematics 125(1-3), 115–122 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockayne, E.J., Hedetniemi, S.T.: Optimal domination in graphs. IEEE Transactions on Circuits and Systems 22, 855–857 (1975)

    Article  MathSciNet  Google Scholar 

  5. Cockayne, E.J., Hedetniemi, S.T.: Towards a theory of domination in graphs. Networks 7, 247–261 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cockayne, E.J., Hedetniemi, S.T.: A linear algorithm for the domination number of a tree. Information Processing Letters 4, 41–44 (1975)

    Article  MATH  Google Scholar 

  7. Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified np-complete graph problems. Theoretical Computer Science 1, 237–267 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of np-completeness (1979)

    Google Scholar 

  10. Hoang, C.: Ph.d. thesis. McGill University, Montreal, Canada (1985)

    Google Scholar 

  11. Kaplan, H., Shamir, R.: The domatic number problem on some perfect graph families. Information Processing Letters 49, 51–56 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lu, T.L., Ho, P.H., Chang, G.J.: The domatic number problem in interval graphs. SIAM Journal on Discrete Mathematics 3, 531–536 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Manacher, G.K., Mankus, T.A.: Finding a domatic partition of an interval graph in time o(n). SIAM Journal on Discrete Mathematics 9, 167–172 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dailey, D.P.: Uniqueness of colorability and colorability of planar 4-regular graphs are np-complete. Discrete Mathematics 30, 289–293 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peng, S.L., Chang, M.S.: A simple linear time algorithm for the domatic partition problem on strongly chordal graphs. Information Processing Letters 43, 297–300 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rao, A.S., Rangan, C.P.: Linear algorithm for domatic number problem on interval graphs. Information Processing Letters 33, 29–33 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rautenbach, D., Volkmann, L.: The domatic number of block-cactus graphs. Discrete Mathematics 187, 185–193 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tinhofer, G.: Strong tree-cographs are birkhoff graphs. Discrete Applied Mathematics 22, 275–288 (1988)

    Article  MathSciNet  Google Scholar 

  19. Tsui, K.W.: On the domatic number of bipartite permutation graphs. Master thesis, Dept. of Computer Science, National Tsing Hua University, Hsinchu, Taiwan, R.O.C (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan

    Sheung-Hung Poon & Chin-Ting Ung

  2. Department of Information Management, Shih Hsin University, Taipei, Taiwan

    William Chung-Kung Yen

Authors
  1. Sheung-Hung Poon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William Chung-Kung Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chin-Ting Ung
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computing Science, University of Alberta, T6G 2E8, Edmonton, Alberta, Canada

    Guohui Lin

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Poon, SH., Yen, W.CK., Ung, CT. (2012). Domatic Partition on Several Classes of Graphs. In: Lin, G. (eds) Combinatorial Optimization and Applications. COCOA 2012. Lecture Notes in Computer Science, vol 7402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31770-5_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31770-5_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31769-9

  • Online ISBN: 978-3-642-31770-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation