Acta Mathematica Sinica, English Series

, Volume 28, Issue 3, pp 463–468

# 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 n 1(T) = ε(G), where n 1(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

05C69

## Supplementary material

10114_2011_66_MOESM1_ESM.tex (26 kb)
Supplementary material, approximately 25.5 KB.
10114_2011_66_MOESM2_ESM.zip (136 kb)
Supplementary material, approximately 135 KB.

## References

1. [1]
Nieminen, J.: Two bounds for the domination number of a graph. Journal of the Institute of Mathematics and its Applications, 14, 183–187 (1974)
2. [2]
Hedetniemi, S. T., Laskar, R. C.: Connected domination in graphs. In: Graph Theory and Combinatorics (B. Bollobás, ed.), Academic Press, London, 1984, 209–218Google Scholar
3. [3]
Duchet, P., Meyniel, H.: On Hadwiger’s number and the stability number. North-Holland Math. Stud., 62, North-Holland, Amsterdam-New York, 1982, 71–73
4. [4]
Lemańska, M.: Lower bound on the domination number of a tree. Discuss. Math. Graph Theory, 24, 165–169 (2004)
5. [5]
Henning, M., Mukwembi, S.: Domination, radius and minimum degree. Discrete Appl. Math., 157, 2964–2968 (2009)
6. [6]
Shan, E. F., Sohn, M. Y., Yuan, X. D., Henning, M. A.: Domination number in graphs with minimum degree two. Acta Mathematica Sinica, English Series, 25(8), 1253–1268 (2009)

© 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