Skip to main content

Complete \(r\)-partite Graphs Determined by their Domination Polynomial

Abstract

The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete \(r\)-partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour et al. (Ars Comb, 2014).

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

Fig. 1
Fig. 2
Fig. 3

References

  1. Aalipour, G., Akbari, S., Ebrahimi, Z.: On \({D}\)-equivalence class of \({K}_{m, n}\). Unpublished manuscript

  2. Aalipour-Hafshejani, G., Akbari, S., Ebrahimi, Z.: On \({D}\)-equivalence class of complete bipartite graphs. Ars Comb. 117, 275–288 (2014)

  3. Akbari, S., Alikhani, S., Peng, Y.H.: Characterization of graphs using domination polynomials. Eur. J. Comb. 31(7), 1714–1724 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. Alikhani, S.: Dominating sets and domination polynomials of graphs. Ph.D. thesis, University Putra Malaysia (2009)

  5. Alikhani, S., Peng, Y.H.: Dominating sets and domination polynomials of certain graphs II. Opusc. Math. 30(1), 37–51 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  6. Alikhani, S., Peng, Y.H.: Domination polynomials of cubic graphs of order 10. Turk. J. Math. 35(3), 355–366 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Alikhani, S., Peng, Y.H.: Introduction to domination polynomial of a graph. Ars Comb. 114, 257–266 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Arocha, J.L., Llano, B.: Mean value for the matching and dominating polynomial. Discus. Math. Graph Theory 20(1), 57–69 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  9. Birkhoff, G.D.: A determinant formula for the number of ways of coloring a map. Ann. Math. 14(1/4), 42–46 (1912)

    MathSciNet  Article  MATH  Google Scholar 

  10. Cameron, P.J.: Research problems from the BCC22. Discret. Math. 311(13), 1074–1083 (2011)

    Article  Google Scholar 

  11. Gutman, I., Harary, F.: Generalizations of the matching polynomial. Utilitas Math. 24, 97–106 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Keevash, P.: Shadows and intersections: stability and new proofs. Adv. Math. 218, 1685–1703 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  13. Lovász, L.: Combinatorial Problems and Exercises. AMS Chelsea Publishing Series. AMS Chelsea Pub, Providence (1993)

    Google Scholar 

  14. Turán, P.: On an extremal problem in graph theory (in Hungarian). Math. Fiz. Lapok 48, 436–452 (1941)

    Google Scholar 

Download references

Acknowledgments

We would like to thank Ghodratollah Aalipour for providing us with preprints of [2] and [1].

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael E. Picollelli.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Anthony, B.M., Picollelli, M.E. Complete \(r\)-partite Graphs Determined by their Domination Polynomial. Graphs and Combinatorics 31, 1993–2002 (2015). https://doi.org/10.1007/s00373-014-1521-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-014-1521-2

Keywords

  • Domination polynomial
  • Dominating set
  • \(\mathcal{D}\)-unique graphs

Mathematics Subject Classification

  • 05C69
  • 05C75