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A proof of a conjecture on the paired-domination subdivision number

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Abstract

A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. It was conjectured that the paired-domination subdivision number is at most \( n-1\) for every connected graph G of order \(n\ge 3\) which does not contain isolated vertices. In this paper, we settle the conjecture in the affirmative.

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References

  1. Amjadi, J., Chellali, M.: Complexity of the paired domination subdivision problem. Commun. Comb. Optim. 7, 177–182 (2022)

    MathSciNet  Google Scholar 

  2. Chellali, M., Haynes, T.W.: Trees with unique minimum paired dominating sets. Ars Combin. 73, 3–12 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Chellali, M., Haynes, T.W.: On paired and double domination in graphs. Util. Math. 67, 161–171 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Egawa, Y., Furuya, M., Takatou, M.: Upper bounds on the paired-domination subdivision number of a graph. Graphs Combin. 29, 843–856 (2013)

    Article  MathSciNet  Google Scholar 

  5. Favaron, O., Karami, H., Sheikholeslami, S.M.: Paired-domination number of a graph and its complement. Discrete Math. 308, 6601–6605 (2008)

    Article  MathSciNet  Google Scholar 

  6. Favaron, O., Karami, H., Sheikholeslami, S.M.: Paired-domination subdivision numbers of graphs. Graphs Combin. 25, 503–512 (2009)

    Article  MathSciNet  Google Scholar 

  7. Hao, G., Sheikholeslami, S.M., Chellali, M., Khoeilar, R., Karami, H.: On the paired-domination subdivision number of a graph, Mathematics, 9, Article ID: 439 (2021)

  8. Haynes, T.W., Henning, M.A.: Paired-domination game played in graphs. Commun. Comb. Optim. 4, 79–94 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Haynes, T.W., Slater, P.J.: Paired-domination in graphs. Networks 32, 199–206 (1998)

    Article  MathSciNet  Google Scholar 

  10. Haynes, T.W., Slater, P.J.: Paired-domination and the paired-domatic number. Congr. Numer. 109, 65–72 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Kang, L., Sohn, M.Y., Cheng, T.C.E.: Paired-domination in inflated graphs. Theor. Comput. Sci. 320, 485–494 (2004)

    Article  MathSciNet  Google Scholar 

  12. Proffitt, K.E., Haynes, T.W., Slater, P.J.: Paired-domination in grid graphs. Congr. Numer. 150, 161–172 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Qiang, X., Kosari, S., Shao, Z., Sheikholeslami, S.M., Chellali, M., Karami, H.: A note on the paired-domination subdivision number of trees. Mathematics, 9, Article ID: 181 (2021)

  14. Wei, S., Hao, G., Sheikholeslami, S.M., Khoeilar, R., Karami, H.: On the paired-domination subdivision number of trees, Mathematics, 9, Article ID: 1135, (2021)

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Acknowledgements

The authors are grateful to anonymous referee for his/her remarks and suggestions that helped improve the manuscript. This work was supported by the Natural Science Foundation of China under Grant 62172116 and the Natural Science Foundation of Guangdong Province under Grant 2021A1515011940.

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Authors and Affiliations

  1. Institute of Computing Science and Technology, Guangzhou University, Guangzhou, 510006, China

    Zehui Shao

  2. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Islamic Republic of Iran

    Seyed Mahmoud Sheikholeslami, Rana Khoeilar & Hossein Karami

  3. LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria

    Mustapha Chellali

Authors
  1. Zehui Shao
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  2. Seyed Mahmoud Sheikholeslami
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  3. Mustapha Chellali
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  4. Rana Khoeilar
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  5. Hossein Karami
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Correspondence to Seyed Mahmoud Sheikholeslami.

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Shao, Z., Sheikholeslami, S.M., Chellali, M. et al. A proof of a conjecture on the paired-domination subdivision number. Graphs and Combinatorics 38, 71 (2022). https://doi.org/10.1007/s00373-022-02472-4

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  • DOI: https://doi.org/10.1007/s00373-022-02472-4

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