Skip to main content
Log in

Subgraph Transversal of Graphs

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ F (G) bounded by a constant fraction of |G|?

In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K k or \(\overline{K_k}\).

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.

Similar content being viewed by others

References

  1. Aigner, M., Andreae, T.: Vertex sets that meet all maximal cliques of a graph, manuscript (1986)

  2. Andreae, T., Schughart, M., Tuza, Z.: Clique-transversal sets of line graphs and complements of line graphs. Discrete Math. 88, 11–20 (1991)

    Google Scholar 

  3. Bacsó, G., Graver, S., Gyárfás, A., Preissmann, M., Sebö, A.: Coloring the maximal cliques of graphs. SIAM J. Discrete Math. 17, 361–376 (2004)

    Google Scholar 

  4. Berge, C.: Hypergraphs. North Holland, New York (1989)

  5. Diestel, R.: Graph Theory, 3rd edn. Springer, New York (2005)

  6. Duffus, D., Sands, B., Winkler, P.: Maximal chains and antichains in the Boolean lattice. SIAM J. Discrete Math. 3, 197–205 (1990)

    Google Scholar 

  7. Duffus, D., Kierstead, H.A., Trotter, W.T.: Fibers and ordered set coloring. J. Combin. Theory Ser. A 58, 158–164 (1991)

    Google Scholar 

  8. Erdös, P., Hajnal, A.: On chromatic numbers of graphs and set systems. Acta Math. Acad. Sci. Hungar. 17, 61–99 (1966)

    Google Scholar 

  9. Erdös, P., Gallai, T., Tuza, Z.: Covering the cliques of a graph with vertices. Discrete Math. 108, 279–289 (1992)

    Google Scholar 

  10. Lonc, Z., Rival, I.: Chains, antichains, and fibres. J. Combin. Theory Ser. A 44, 207–228 (1987)

    Google Scholar 

  11. Nešetřil, J., Rödl, V.: Partitions of vertices. Comm. Math. Univ. Carolinae 17, 85–95 (1976)

    Google Scholar 

  12. Tuza, Z.: Covering all cliques of a graph. Discrete Math. 86, 117–126 (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jia Shen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, J. Subgraph Transversal of Graphs. Graphs and Combinatorics 25, 601–609 (2009). https://doi.org/10.1007/s00373-009-0850-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-009-0850-z

Keywords

Navigation