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Graphs and Combinatorics

, Volume 28, Issue 2, pp 199–214 | Cite as

Girth and Total Domination in Graphs

  • Michael A. Henning
  • Anders Yeo
Original Paper

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. The girth of G is the length of a shortest cycle in G. Let G be a connected graph with minimum degree at least 2, order n and girth g ≥ 3. It was shown in an earlier manuscript (Henning and Yeo in Graphs Combin 24:333–348, 2008) that \({\gamma_t(G)\le(\frac{1}{2}+\frac{1}{g})n}\), and this bound is sharp for cycles of length congruent to two modulo four. In this paper we show that \({\gamma_t(G)\le\frac{n}{2}+\max(1,\frac{n}{2(g+1)})}\), and this bound is sharp.

Keywords

Girth Total domination 

Mathematics Subject Classification (2000)

05C69 

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Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  2. 2.Department of Computer Science, Royal HollowayUniversity of LondonEgham, SurreyUK

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