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On the Total Domination Number of Cartesian Products of Graphs

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Abstract.

The most famous open problem involving domination in graphs is Vizing’s conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.

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Authors and Affiliations

  1. School of Mathematics, Statistics, & Information Technology, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa

    Michael A. Henning

  2. Department of Mathemtics, Furman University, Greenville, SC, 29613, USA

    Douglas F. Rall

Authors
  1. Michael A. Henning
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  2. Douglas F. Rall
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Correspondence to Michael A. Henning.

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Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal

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Henning, M., Rall, D. On the Total Domination Number of Cartesian Products of Graphs. Graphs and Combinatorics 21, 63–69 (2005). https://doi.org/10.1007/s00373-004-0586-8

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  • DOI: https://doi.org/10.1007/s00373-004-0586-8

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