The limit case of a domination property

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, ...}.