Graphs with least domination number three-fifths their order Original Research Article

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 γℓ(G) is the minimum cardinality of a least dominating set. Sampathkumar (Discrete Math. 86 (1990) 137–142) conjectured that γℓ(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 γℓ(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 γℓ(G)=3n/5.