Vertex domination-critical graphs

A graph G is vertex domination-critical if for any vertex v of G the domination number of G - v is less than the domination number of G. If such a graph G has domination number γ, it is called γ-critical. Brigham et al. studied γ-critical graphs and posed the following questions: (1) If G is a γ-critical graph, is |V| ≥ (δ + 1)(γ - 1) + 1?(2) If a γ-critical graph G has (Δ + 1)(γ - 1) + 1 vertices, is G regular? (3) Does i = γ for all γ-critical graphs? (4) Let d be the diameter of the γ-critical graph G. Does d ≤ 2(γ - 1) always hold? We show that the first and third questions have a negative answer and the others have a positive answer.