Graphs and Combinatorics

, Volume 35, Issue 1, pp 103–118

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

• Eglantine Camby
Original Paper

## 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.

## Keywords

Vertex cover Connected vertex cover Dominating set Connected dominating set Forbidden induced subgraph Extremal graph

## Notes

### Acknowledgements

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

## References

1. 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)Google Scholar
2. 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)
3. 3.
Camby, E., Caporossi, G.: Studying graphs and their induced subgraphs with the computer: GraphsInGraphs. Cahiers du GERAD G-2016-10 (2016)Google Scholar
4. 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)
5. 5.
Camby, E., Plein, F.: A note on an induced subgraph characterization of domination perfect graphs. Discret. Appl. Math. 217, 711–717 (2017)
6. 6.
Camby, E., Schaudt, O.: The price of connectivity for dominating sets: upper bounds and complexity. Discret. Appl. Math. 177, 53–59 (2014)
7. 7.
Cardinal, J., Levy, E.: Connected vertex covers in dense graphs. Theor. Comput. Sci. 411(26–28), 2581–2590 (2010)
8. 8.
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer-Verlag, Heidelberg (2005)Google Scholar
9. 9.
Duchet, P., Meyniel, H.: On Hadwiger’s number and the stability number. Ann. Discret. Math. 13, 71–74 (1982)
10. 10.
Fulman, J.: A note on the characterization of domination perfect graphs. J. Graph Theory 17, 47–51 (1993)
11. 11.
Hartinger, T.R., Johnson, M., Milanič, M., Paulusma, D.: The price of connectivity for cycle transversals. Eur. J. Comb. 58, 203–224 (2016)
12. 12.
Levy, E.: Approximation algorithms for covering problems in dense graphs. Ph.D. thesis, Université libre de Bruxelles, Brussels (2009)Google Scholar
13. 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)
14. 14.
Sumner, D.P., Moore, J.I.: Domination perfect graphs. Not. Am. Math. Soc. 26, A–569 (1979)Google Scholar
15. 15.
Tuza, Z.: Hereditary domination in graphs: characterization with forbidden induced subgraphs. SIAM J. Discret. Math. 22, 849–853 (2008)
16. 16.
Zverovich, I.E.: Perfect connected-dominant graphs. Discuss. Math. Graph Theory 23, 159–162 (2003)
17. 17.
Zverovich, I.E., Zverovich, V.E.: A characterization of domination perfect graphs. J. Graph Theory 15, 109–114 (1991)
18. 18.
Zverovich, I.E., Zverovich, V.E.: An induced subgraph characterization of domination perfect graphs. J. Graph Theory 20, 375–395 (1995)
19. 19.
Zverovich, I.E., Zverovich, V.E.: A semi-induced subgraph characterization of upper domination perfect graphs. J. Graph Theory 31, 29–49 (1999)

© Springer Japan KK, part of Springer Nature 2019

## Authors and Affiliations

• Eglantine Camby
• 1
• 2
• 3
1. 1.Université Libre de BruxellesBrusselsBelgium