In the paper [1], the following problems have been proposed. Is it true that every connected bicritical graph has a minimum dominating set containing any two specified vertices of the graphs? Is it true if G is a connected bicritical graph, then γ(G) = i(G), where i(G) is the independent domination number? We disprove the second problem and show the truth of the first problem for a certain family of graphs. Furthermore this family of graphs is characterized with respect to bicriticality, diameter, vertex connectivity and edge connectivity.
