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

Graphs and Combinatorics volume 31, pages 1993–2002 (2015)Cite this article

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).

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Acknowledgments

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

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Authors and Affiliations

  1. Southwestern University, 1001 E. University Ave, Georgetown, TX, 78626, USA

    Barbara M. Anthony

  2. California State University San Marcos, San Marcos, CA, 92096, USA

    Michael E. Picollelli

Authors
  1. Barbara M. Anthony
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  2. Michael E. Picollelli
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Correspondence to Michael E. Picollelli.

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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

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

Keywords

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

Mathematics Subject Classification

  • 05C69
  • 05C75