Graphs with least domination number three-fifths their order

Let G = (V,E) be a graph. A set S ⊆ V is a dominating set if every vertex of V - S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. A dominating set D is a least dominating set if γ(〈D〉) ≤ γ(〈S〉) for any dominating set S, and γ_ℓ(G) is the minimum cardinality of a least dominating set. Sampathkumar (Discrete Math. 86 (1990) 137-142) conjectured that γ_ℓ(G) < 3n/5 for every connected graph on n ≥ 2 vertices. This conjecture was proven by Favaron (Discrete Math. 150 (1996) 115-122). We shall characterise graphs G of order n that are edge-minimal with respect to satisfying G connected and γ_ℓ(G) = 3n/5. Furthermore, we construct a family of graphs G of order n that are not cycles and are edge-minimal with respect to satisfying G connected, δ(G) ≥ 2 and γ_ℓ(G) = 3n/5.