Graphs and Combinatorics

, Volume 31, Issue 6, pp 1993–2002 | Cite as

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

Original Paper


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).


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

Mathematics Subject Classification

05C69 05C75 



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


  1. 1.
    Aalipour, G., Akbari, S., Ebrahimi, Z.: On \({D}\)-equivalence class of \({K}_{m, n}\). Unpublished manuscriptGoogle Scholar
  2. 2.
    Aalipour-Hafshejani, G., Akbari, S., Ebrahimi, Z.: On \({D}\)-equivalence class of complete bipartite graphs. Ars Comb. 117, 275–288 (2014)Google Scholar
  3. 3.
    Akbari, S., Alikhani, S., Peng, Y.H.: Characterization of graphs using domination polynomials. Eur. J. Comb. 31(7), 1714–1724 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alikhani, S.: Dominating sets and domination polynomials of graphs. Ph.D. thesis, University Putra Malaysia (2009)Google Scholar
  5. 5.
    Alikhani, S., Peng, Y.H.: Dominating sets and domination polynomials of certain graphs II. Opusc. Math. 30(1), 37–51 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Alikhani, S., Peng, Y.H.: Domination polynomials of cubic graphs of order 10. Turk. J. Math. 35(3), 355–366 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Alikhani, S., Peng, Y.H.: Introduction to domination polynomial of a graph. Ars Comb. 114, 257–266 (2014)MathSciNetMATHGoogle Scholar
  8. 8.
    Arocha, J.L., Llano, B.: Mean value for the matching and dominating polynomial. Discus. Math. Graph Theory 20(1), 57–69 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Birkhoff, G.D.: A determinant formula for the number of ways of coloring a map. Ann. Math. 14(1/4), 42–46 (1912)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cameron, P.J.: Research problems from the BCC22. Discret. Math. 311(13), 1074–1083 (2011)CrossRefGoogle Scholar
  11. 11.
    Gutman, I., Harary, F.: Generalizations of the matching polynomial. Utilitas Math. 24, 97–106 (1983)MathSciNetMATHGoogle Scholar
  12. 12.
    Keevash, P.: Shadows and intersections: stability and new proofs. Adv. Math. 218, 1685–1703 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lovász, L.: Combinatorial Problems and Exercises. AMS Chelsea Publishing Series. AMS Chelsea Pub, Providence (1993)Google Scholar
  14. 14.
    Turán, P.: On an extremal problem in graph theory (in Hungarian). Math. Fiz. Lapok 48, 436–452 (1941)Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Southwestern University1001 E. University AveGeorgetownUSA
  2. 2.California State University San MarcosSan MarcosUSA

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