Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Combinatorial Algorithms

IWOCA 2019: Combinatorial Algorithms pp 352–364Cite as

Complexity of Fall Coloring for Restricted Graph Classes

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11638))

Abstract

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330–338) by proving that it is \(\textsc {NP}\)-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into k independent dominating sets is \(\textsc {NP}\)-complete for every \(k \ge 3\). We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451–6458) by showing that it is \(\textsc {NP}\)-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every \(k \ge 3\), we show that there is some constant t depending only on k such that deciding whether a k-regular graph can be partitioned into t independent dominating sets is \(\textsc {NP}\)-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

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   59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   79.99
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. de Berg, M., Khosravi, A.: Optimal binary space partitions for segments in the plane. Int. J. Comput. Geom. Appl. 22(03), 187–205 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chartrand, G., Geller, D.P.: On uniquely colorable planar graphs. J. Comb. Theory 6(3), 271–278 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  8. Dunbar, J.E., et al.: Fall colorings of graphs. J. Comb. Math. Comb. Comput. 33, 257–274 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Fomin, F.V., Thilikos, D.M.: New upper bounds on the decomposability of planar graphs. J. Graph Theory 51(1), 53–81 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313(7), 839–854 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heggernes, P., Telle, J.A.: Partitioning graphs into generalized dominating sets. Nord. J. Comput. 5(2), 128–142 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Henning, M.A., Löwenstein, C., Rautenbach, D.: Remarks about disjoint dominating sets. Discrete Math. 309(23), 6451–6458 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Laskar, R., Lyle, J.: Fall colouring of bipartite graphs and cartesian products of graphs. Discrete Appl. Math. 157(2), 330–338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Leven, D., Galil, Z.: NP-completeness of finding the chromatic index of regular graphs. J. Algorithms 4(1), 35–44 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lewis, R.: A Guide to Graph Colouring. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25730-3

    Book  Google Scholar 

  17. Maffray, F., Preissmann, M.: On the NP-completeness of the \(k\)-colorability problem for triangle-free graphs. Discrete Math. 162(1–3), 313–317 (1996)

    MathSciNet  MATH  Google Scholar 

  18. Marx, D.: Graph colouring problems and their applications in scheduling. Electr. Eng. 48(1–2), 11–16 (2004)

    Google Scholar 

  19. Mitillos, C.: Topics in graph fall-coloring. Ph.D. thesis, Illinois Institute of Technology (2016)

    Google Scholar 

  20. Resende, M.G., Pardalos, P.M.: Handbook of Optimization in Telecommunications. Springer, Boston (2008). https://doi.org/10.1007/978-0-387-30165-5

    Book  MATH  Google Scholar 

  21. van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic programming on tree decompositions using generalised fast subset convolution. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 566–577. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04128-0_51

    Chapter  Google Scholar 

  22. Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math. 10, 529–550 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nokia Bell Labs, Dublin, Ireland

    Juho Lauri

  2. Illinois Institute of Technology, Chicago, USA

    Christodoulos Mitillos

Authors
  1. Juho Lauri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christodoulos Mitillos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juho Lauri .

Editor information

Editors and Affiliations

  1. Arizona State University, Tempe, AZ, USA

    Charles J. Colbourn

  2. University of Pisa, Pisa, Italy

    Roberto Grossi

  3. University of Pisa, Pisa, Italy

    Nadia Pisanti

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

Lauri, J., Mitillos, C. (2019). Complexity of Fall Coloring for Restricted Graph Classes. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-25005-8_29

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-25004-1

  • Online ISBN: 978-3-030-25005-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation