Common extremal graphs for three inequalities involving domination parameters

‎Let \delta (G)‎, ‎\Delta (G) and \gamma(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) \leq \delta(G)‎ + ‎۱‎, ‎d(G)\gamma(G) \leq |V(G)| \cite{ch}‎, ‎and |V(G)| \leq (\Delta(G)‎+‎۱)\gamma(G) \cite{berge}‎. ‎In this paper‎, ‎we investigate the graphs G for which‎ ‎all the above inequalities become simultaneously equalities‎.

Keywords:

‎domination/domatic/idomatic number‎ , ‎efficient dominating set