On the domination number of prisms of graphs

For a permutation π of the vertex set of a graph G, the graph πG is obtained from two disjoint copies G1 and G2 of G by joining each v in G1 to π(v) in G2. Hence if π = 1, then πG = K2 × G, the prism of G. Clearly, γ(G) ≤ γ(πG) ≤ 2γ(G). We study graphs for which γ(K2 × G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V (G) and those for which γ(πG) = 2γ(G) for each permutation π of V (G).