Skip to main content
View expanded cover

International Conference on Combinatorial Optimization and Applications

COCOA 2014: Combinatorial Optimization and Applications pp 258–267Cite as

A Dichotomy for Upper Domination in Monogenic Classes

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8881)

Abstract

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is NP-hard for general graphs and in many restricted graph families. In the present paper, we study the computational complexity of this problem in monogenic classes of graphs (i.e. classes defined by a single forbidden induced subgraph) and show that the problem admits a dichotomy in this family. In particular, we prove that if the only forbidden induced subgraph is a \(P_4\) or a \(2K_2\) (or any induced subgraph of these graphs), then the problem can be solved in polynomial time. Otherwise, it is NP-hard.

Keywords

  • Monogenic Classes
  • Restricted Graph Families
  • Maximum Cardinality
  • Private Neighbor
  • Polynomial-time Result

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vadim Lozin: The author gratefully acknowledges support from DIMAP - the Center for Discrete Mathematics and its Applications at the University of Warwick, and from EPSRC, grant EP/L020408/1.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-12691-3_20
  • Chapter length: 10 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-12691-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1.

References

  1. Brandstädt, A., Engelfriet, J., Le, H.-O., Lozin, V.V.: Clique-width for 4-vertex forbidden subgraphs. Theory Comput. Syst. 39(4), 561–590 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Cheston, G.A., Fricke, G., Hedetniemi, S.T., Jacobs, D.P.: On the computational complexity of upper fractional domination. Discrete Appl. Math. 27(3), 195–207 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Cockayne, E.J., Favaron, O., Payan, C., Thomason, A.G.: Contributions to the theory of domination, independence and irredundance in graphs. Discrete Math. 33(3), 249–258 (1981)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. 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 

  6. Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-Complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Hare, E.O., Hedetniemi, S.T., Laskar, R.C., Peters, K., Wimer, T.: Linear-time computability of combinatorial problems on generalized-series-parallel graphs. In: Johnson, D.S., et al. (eds.) Discrete Algorithms and Complexity, pp. 437–457. Academic Press, New York (1987)

    CrossRef  Google Scholar 

  8. Jacobson, M.S., Peters, K.: Chordal graphs and upper irredundance, upper domination and independence. Discrete Math. 86(1–3), 59–69 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Kamiński, M.: MAX-CUT and containment relations in graphs. Theor. Comput. Sci. 438, 89–95 (2012)

    CrossRef  MATH  Google Scholar 

  10. Korobitsyn, D.V.: On the complexity of determining the domination number in monogenic classes of graphs. Diskretnaya Matematika 2(3), 90–96 (1990)

    MATH  Google Scholar 

  11. Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of Coloring Graphs without Forbidden Induced Subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  12. Lozin, V.V., Mosca, R.: Independent sets in extensions of \(2K_2\)-free graphs. Discrete Appl. Math. 146(1), 74–80 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. King Abdullah University of Science and Technology, Thuwal, Saudia Arabia

    Hassan AbouEisha & Shahid Hussain

  2. DIMAP and Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Vadim Lozin

  3. PSL, Université Paris-Dauphine, 75775, Paris Cedex 16, France

    Jérôme Monnot & Bernard Ries

  4. CNRS, LAMSADE UMR 7243, Paris, France

    Jérôme Monnot & Bernard Ries

Authors
  1. Hassan AbouEisha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shahid Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vadim Lozin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jérôme Monnot
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bernard Ries
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vadim Lozin .

Editor information

Editors and Affiliations

  1. Zhejiang Normal University, Jinhua, Zhejiang, China

    Zhao Zhang

  2. University of Texas, Tyler, Texas, USA

    Lidong Wu

  3. University of Texas, Dallas, Texas, USA

    Wen Xu

  4. University of Texas, Dallas, USA

    Ding-Zhu Du

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

AbouEisha, H., Hussain, S., Lozin, V., Monnot, J., Ries, B. (2014). A Dichotomy for Upper Domination in Monogenic Classes. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-12691-3_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12690-6

  • Online ISBN: 978-3-319-12691-3

  • eBook Packages: Computer ScienceComputer Science (R0)