Domination, radius, and minimum degree

We prove sharp bounds concerning domination number, radius, order and minimum degree of a graph. In particular, we prove that if GG is a connected graph of order nn, domination number γγ and radius rr, then 23r≤γ≤n−43r+23. Equality is achieved in the upper bound if, and only if, GG is a path or a cycle on nn vertices with n≡4(mod6)n≡4(mod6). Further, if GG has minimum degree δ≥3δ≥3 and r≥6r≥6, then using a result due to Erdös, Pach, Pollack, and Tuza [P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree. J. Combin. Theory B 47 (1989), 73–79] we show that γ≤n−23(r−6)δ.