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Total Domination in Graphs with Given Girth

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Abstract

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ t (G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g, then γ t (G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs.

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Authors and Affiliations

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

    Michael A. Henning

  2. Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 OEX, UK

    Anders Yeo

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  1. Michael A. Henning
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  2. Anders Yeo
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Michael A. Henning: Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.

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Henning, M.A., Yeo, A. Total Domination in Graphs with Given Girth. Graphs and Combinatorics 24, 333–348 (2008). https://doi.org/10.1007/s00373-008-0797-5

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  • DOI: https://doi.org/10.1007/s00373-008-0797-5

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