In the paper [R.C. Brigham, et. al. Bicritical domination, Discrete Math, 305 (2005) 18-32] the problem; is it true that every connected bicritical graph has a minimum dominating set containing any two specified vertices of the graphs? We give a class of graphs that disprove this problem and furthermore we obtain the domination numbers and the diameters of the graphs of this class. This class of graphs has the property: γ(G) − diam(G) → ∞ when ∣V (G)∣ = n → ∞. Also for the bicritical graphs of this class i(G) = γ(G).
