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Graphs and Combinatorics

, Volume 24, Issue 4, pp 333–348 | Cite as

Total Domination in Graphs with Given Girth

  • Michael A. Henning
  • Anders Yeo
Article

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.

Keywords

Graphs Hypergraphs Total domination number Transversals Girth 

AMS subject classification

05C70 

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Copyright information

© Springer Japan 2008

Authors and Affiliations

  • Michael A. Henning
    • 1
  • Anders Yeo
    • 2
  1. 1.School of Mathematical SciencesUniversity of KwaZulu-NatalPietermaritzburgSouth Africa
  2. 2.Department of Computer Science, Royal HollowayUniversity of LondonSurreyUK

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