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On the upper total domination number of Cartesian products of graphs

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Abstract

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63–69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ t (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γ t (G)Γ t (H)≤2Γ t (G □ H).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa

    Paul Dorbec & Michael A. Henning

  2. Institut Fourier Organization UJF-ERTé “Maths à modeler”, 100 rue des Maths, 38402, St Martin d’Hères, France

    Paul Dorbec

  3. Department of Mathematics, Furman University, Greenville, SC, 29613, USA

    Douglas F. Rall

Authors
  1. Paul Dorbec
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  2. Michael A. Henning
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  3. Douglas F. Rall
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Corresponding author

Correspondence to Paul Dorbec.

Additional information

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

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Dorbec, P., Henning, M.A. & Rall, D.F. On the upper total domination number of Cartesian products of graphs. J Comb Optim 16, 68–80 (2008). https://doi.org/10.1007/s10878-007-9099-8

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  • DOI: https://doi.org/10.1007/s10878-007-9099-8

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