Skip to main content
Log in

Price of Connectivity for the Vertex Cover Problem and the Dominating Set Problem: Conjectures and Investigation of Critical Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The vertex cover problem and the dominating set problem are two well-known problems in graph theory. Their goal is to find the minimum size of a vertex subset satisfying some properties. Both hold a connected version, which imposes that the vertex subset must induce a connected component. To study the interdependence between the connected version and the original version of a problem, the Price of Connectivity (\(PoC\)) was introduced by Cardinal and Levy (Theor Comput Sci 411(26–28):2581–2590, 2010) and Levy (Approximation algorithms for covering problems in dense graphs. Ph.D. thesis, Université libre de Bruxelles, Brussels, 2009) as the ratio between invariants from the connected version and the original version of the problem. Camby et al. (Discret Math Theor Comput Sci 16:207–224, 2014) for the vertex cover problem, Camby and Schaudt (Discret Appl Math 177:53–59, 2014) for the dominating set problem characterized some classes of \(PoC\)-Near-Perfect graphs, hereditary classes of graphs in which the Price of Connectivity is bounded by a fixed constant. Moreover, only for the vertex cover problem, Camby et al. (2014) introduced the notion of critical graphs, graphs that can appear in the list of forbidden induced subgraphs characterization. By definition, the Price of Connectivity of a critical graph is strictly greater than that of any proper induced subgraph. In this paper, we prove that for the vertex cover problem, every critical graph is either isomorphic to a cycle on 5 vertices or bipartite. To go further in the previous studies, we also present conjectures on \(PoC\)-Near-Perfect graphs and critical graphs with the help of the computer software GraphsInGraphs (Camby and Caporossi in Studying graphs and their induced subgraphs with the computer: GraphsInGraphs. Cahiers du GERAD G-2016-10, 2016). Moreover, for the dominating set problem, we investigate critical trees and we show that every minimum dominating set of a critical graph is independent.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Belmonte, R., van ’t Hof, P., Kamiński, M., Paulusma, D.: Forbidden induced subgraphs and the price of connectivity for feedback vertex set. In: Csuhaj-Varjú E., Dietzfelbinger M., Ésik Z. (eds.) International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 8635, pp. 57–68. Springer, Berlin, Heidelberg (2014)

  2. Belmonte, R., van ’t Hof, P., Kamiński, M., Paulusma, D.: The price of connectivity for feedback vertex set. Discret. Appl. Math. 217, 132–143 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Camby, E., Caporossi, G.: Studying graphs and their induced subgraphs with the computer: GraphsInGraphs. Cahiers du GERAD G-2016-10 (2016)

  4. Camby, E., Cardinal, J., Fiorini, S., Schaudt, O.: The price of connectivity for vertex cover. Discret. Math. Theor. Comput. Sci. 16, 207–224 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Camby, E., Plein, F.: A note on an induced subgraph characterization of domination perfect graphs. Discret. Appl. Math. 217, 711–717 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Camby, E., Schaudt, O.: The price of connectivity for dominating sets: upper bounds and complexity. Discret. Appl. Math. 177, 53–59 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cardinal, J., Levy, E.: Connected vertex covers in dense graphs. Theor. Comput. Sci. 411(26–28), 2581–2590 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer-Verlag, Heidelberg (2005)

  9. Duchet, P., Meyniel, H.: On Hadwiger’s number and the stability number. Ann. Discret. Math. 13, 71–74 (1982)

    MathSciNet  MATH  Google Scholar 

  10. Fulman, J.: A note on the characterization of domination perfect graphs. J. Graph Theory 17, 47–51 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hartinger, T.R., Johnson, M., Milanič, M., Paulusma, D.: The price of connectivity for cycle transversals. Eur. J. Comb. 58, 203–224 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Levy, E.: Approximation algorithms for covering problems in dense graphs. Ph.D. thesis, Université libre de Bruxelles, Brussels (2009)

  13. Schaudt, O.: On graphs for which the connected domination number is at most the total domination number. Discret. Appl. Math. 160, 1281–1284 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sumner, D.P., Moore, J.I.: Domination perfect graphs. Not. Am. Math. Soc. 26, A–569 (1979)

  15. Tuza, Z.: Hereditary domination in graphs: characterization with forbidden induced subgraphs. SIAM J. Discret. Math. 22, 849–853 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zverovich, I.E.: Perfect connected-dominant graphs. Discuss. Math. Graph Theory 23, 159–162 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zverovich, I.E., Zverovich, V.E.: A characterization of domination perfect graphs. J. Graph Theory 15, 109–114 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zverovich, I.E., Zverovich, V.E.: An induced subgraph characterization of domination perfect graphs. J. Graph Theory 20, 375–395 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zverovich, I.E., Zverovich, V.E.: A semi-induced subgraph characterization of upper domination perfect graphs. J. Graph Theory 31, 29–49 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by a post-doc grant “Bourse d’Excellence WBI.WORLD” from Fédération Wallonie-Bruxelles (Belgium).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eglantine Camby.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Camby, E. Price of Connectivity for the Vertex Cover Problem and the Dominating Set Problem: Conjectures and Investigation of Critical Graphs. Graphs and Combinatorics 35, 103–118 (2019). https://doi.org/10.1007/s00373-018-1981-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-018-1981-x

Keywords

Navigation