Skip to main content
SpringerLink
Log in
Book cover

International Conference on Algorithmic Applications in Management

AAIM 2019: Algorithmic Aspects in Information and Management pp 304–315Cite as

Improved Parameterized Algorithms for Mixed Domination

  • Conference paper
  • First Online:

  • 550 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11640))

Abstract

A mixed domination of a graph \(G = (V, E)\) is a mixed set D of vertices and edges such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The Mixed Domination problem is to check whether there is a mixed domination of size at most k in a graph. Mixed domination is a mixture concept of vertex domination and edge domination, and the mixed domination problem has been studied from the view of approximation algorithms, parameterized algorithms, and so on. In this paper, we give a branch-and-search algorithm with running time bound of \(O^*(4.172^k)\), which improves the previous bound of \(O^*(7.465^k)\). For kernelization, it is known that the problem parameterized by k in general graphs is unlikely to have a polynomial kernel. We show the problem in planar graphs allows linear kernels by giving a kernel of 11k vertices.

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 EPUB and 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

References

  1. Alavi, Y., Behzad, M., Lesniak-Foster, L.M., Nordhaus, E.: Total matchings and total coverings of graphs. J. Graph Theory 1(2), 135–140 (1977)

    Article  MathSciNet  Google Scholar 

  2. Fernau, H.: edge dominating set: efficient enumeration-based exact algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 142–153. Springer, Heidelberg (2006). https://doi.org/10.1007/11847250_13

    Chapter  MATH  Google Scholar 

  3. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16533-7

    Book  Google Scholar 

  4. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    MATH  Google Scholar 

  5. Hatami, P.: An approximation algorithm for the total covering problem. Discussiones Mathematicae Graph Theory 27(3), 553–558 (2007)

    Article  MathSciNet  Google Scholar 

  6. Hedetniemi, S.M., Hedetniemi, S.T., Laskar, R., McRae, A., Majumdar, A.: Domination, independence and irredundance in total graphs: a brief survey. In: Graph Theory, Combinatorics and Applications: Proceedings of the 7th Quadrennial International Conference on the Theory and Applications of Graphs, vol. 2, pp. 671–683. Wiley, New York (1995)

    Google Scholar 

  7. Jain, P., Jayakrishnan, M., Panolan, F., Sahu, A.: Mixed Dominating Set: a parameterized perspective. In: Bodlaender, H.L., Woeginger, G.J. (eds.) WG 2017. LNCS, vol. 10520, pp. 330–343. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68705-6_25

    Chapter  Google Scholar 

  8. Lan, J.K., Chang, G.J.: On the mixed domination problem in graphs. Theoret. Comput. Sci. 476, 84–93 (2013)

    Article  MathSciNet  Google Scholar 

  9. Manlove, D.F.: On the algorithmic complexity of twelve covering and independence parameters of graphs. Discrete Appl. Math. 91(1–3), 155–175 (1999)

    Article  MathSciNet  Google Scholar 

  10. Rajaati, M., Hooshmandasl, M.R., Dinneen, M.J., Shakiba, A.: On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width. Discrete Math. Theoret. Comput. Sci. 20(2), 1–25 (2018)

    Google Scholar 

  11. Xiao, M.: Upper and lower bounds on approximating weighted mixed domination. In: COCOON 2019 (2019, to appear)

    Google Scholar 

  12. Xiao, M., Kloks, T., Poon, S.H.: New parameterized algorithms for the edge dominating set problem. Theoret. Comput. Sci. 511, 147–158 (2013)

    Article  MathSciNet  Google Scholar 

  13. Zhao, Y., Kang, L., Sohn, M.Y.: The algorithmic complexity of mixed domination in graphs. Theoret. Comput. Sci. 412(22), 2387–2392 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China, under grants 61772115 and 61370071.

Author information

Authors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Mingyu Xiao & Zimo Sheng

Authors
  1. Mingyu Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zimo Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingyu Xiao .

Editor information

Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Hefei University of Technology, Hefei, China

    Lian Li

  3. Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun

  4. Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

    Jialin Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Xiao, M., Sheng, Z. (2019). Improved Parameterized Algorithms for Mixed Domination. In: Du, DZ., Li, L., Sun, X., Zhang, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2019. Lecture Notes in Computer Science(), vol 11640. Springer, Cham. https://doi.org/10.1007/978-3-030-27195-4_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-27195-4_28

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-27194-7

  • Online ISBN: 978-3-030-27195-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation