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Graphs and Combinatorics

, Volume 25, Issue 1, pp 41–47 | Cite as

A New Bound on the Total Domination Subdivision Number

  • O. Favaron
  • H. Karami
  • R. Khoeilar
  • S. M. Sheikholeslami
Article

Abstract

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number \(sd_{\gamma_{t}}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for every simple connected graph G of order n ≥ 3,
$${\rm sd}_{\gamma_{t}}(G)\le 3 +{\rm min}\{d_2(v); v\in V \, {\rm and}\, d(v)\ge 2\}$$
where d 2(v) is the number of vertices of G at distance 2 from v.

Keywords

Total domination number Total domination subdivision number 

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Copyright information

© Springer Japan 2009

Authors and Affiliations

  • O. Favaron
    • 1
  • H. Karami
    • 2
  • R. Khoeilar
    • 2
  • S. M. Sheikholeslami
    • 2
  1. 1.Univ Paris-Sud, LRI, UMR 8623OrsayFrance
  2. 2.Department of MathematicsAzarbaijan University of Tarbiat MoallemTabrizI.R. Iran

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