2-Complement of Doubly Connected Domination

If the dominating set D of a graph G = (V, E) is a doubly connected dominating set (dcd-set), then < V/D > and < D > are connected. The minimum cardinality of the dcd-set in G is the dcd number, and it is represented by γ_cc(G). Assuming that P = (W1, W2) is the partition of V from an edge set E(G), eliminate the edges between W2 and W1 in G and join the non-adjacent edges between W1 and W2 that are in G. The graph G_2^P thus obtained is called the 2-complement of G with respect to rPr. The minimum cardinality of a 2-complement of a dcd-set is called the 2-complement of dcd number of G and is represent- ed by γ_cc(G_2^P). In this document, bounds for γ_cc(G_2^P) and exact values of some stand graphs are also found.