# The limit case of a domination property

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## 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## MR(2000) Subject Classification

05C69## Preview

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## Supplementary material

## References

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