Skip to main content
Log in

Most unbreakable murky graphs are bull-free

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

A bull is the (self-complementary) graph with nodesa, b, c, d, e and edgesab, ac, bc, bd,ce. A star cutset in a graph G is a set C of nodes such thatG-C is disconnected, and such that some node inC is adjacent to all remaining nodes inC. A graph is called unbreakable if it has more than two nodes and if neither the graph nor its complement has a star cutset.

Hayward defined a graphG to be murky if neitherG nor its complement contains a chordless cycle with five nodes or a chordless path with six nodes. We prove that most unbreakable murky graphs are bull-free. This leads, via a result of Chvátal and Sbihi, to a shortening of Hayward’s proof that murky graphs are perfect.

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

Similar content being viewed by others

References

  1. Chvátal, V. (1985): Star Cutsets and Perfect Graphs, J. Comb. Theory Sir.B 39, 189–199

    Article  MATH  Google Scholar 

  2. Chvátal, V., N. Sbihi, (1987): Bull-free Berge graphs are perfect, Graphs and Combinatorics3, 127–139

    Article  MATH  MathSciNet  Google Scholar 

  3. Hayward, R.B. (1990): Murky graphs, J. Comb. Theory Sir.B 49, 200–235

    Article  MATH  MathSciNet  Google Scholar 

  4. König, D. (1916): Uber Graphen und ihre Anwendung auf Determinanten-Theorie und Mengenlehre, Math. Ann.77, 453–465

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hertz, A. Most unbreakable murky graphs are bull-free. Graphs and Combinatorics 9, 173–175 (1993). https://doi.org/10.1007/BF02988303

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02988303

Keywords

Navigation