Common Extremal Graphs for Three Inequalities Involving Domination Parameters

Let δ(G), ∆(G) and γ(G) be the minimum degree, maximum degree and domination number of a graph G = (V (G), E(G)), respectively. A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G, denoted d(G). It is well known that d(G) ≤ δ(G)+1, d(G)γ(G) ≤ |V (G)| [6], and |V (G)| ≤ (∆(G) + 1)γ(G) [3]. In this paper, we investigate the graphs G for which all the above inequalities become simultaneously equalities.