Acta Mathematica Sinica, English Series

, Volume 28, Issue 3, pp 463–468 | Cite as

The limit case of a domination property

  • Magdalena Lemańska
  • Juan A. Rodríguez-Velázquez
  • Ismael G. Yero
Article

Abstract

The domination number Γ(G) of a connected graph G of order n is bounded below by \(\tfrac{{n + 2 - \varepsilon (G)}} {3}\), where ε(G) denotes the maximum number of leaves in any spanning tree of G. We show that \(\tfrac{{n + 2 - \varepsilon (G)}} {3} = \gamma (G)\) if and only if there exists a tree \(T \in \mathcal{T}(G) \cap \mathcal{R}\) such that n1(T) = ε(G), where n1(T) denotes the number of leaves of T, \(\mathcal{R}\) denotes the family of all trees in which the distance between any two distinct leaves is congruent to 2 modulo 3, and \(\mathcal{T}\) (G) denotes the set composed by the spanning trees of G. As a consequence of the study, we show that if \(\tfrac{{n + 2 - \varepsilon (G)}} {3} = \gamma (G)\), then there exists a minimum dominating set in G whose induced subgraph is an independent set. Finally, we characterize all unicyclic graphs G for which equality \(\tfrac{{n + 2 - \varepsilon (G)}} {3} = \gamma (G)\) holds and we show that the length of the unique cycle of any unicyclic graph G with \(\tfrac{{n + 2 - \varepsilon (G)}} {3} = \gamma (G)\) belongs to {4} ∪ {3, 6, 9, ...}.

Keywords

Dominating set domination number 

MR(2000) Subject Classification

05C69 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Magdalena Lemańska
    • 1
  • Juan A. Rodríguez-Velázquez
    • 2
  • Ismael G. Yero
    • 2
  1. 1.Department of Technical Physics and Applied MathematicsGdańsk University of TechnologyGdanskPoland
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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