D− Complement and D(i)− Complement of a Graph

A dominating set for a graph G = (V,E) is a subset D of V such that every point not in D is adjacent to at least one member of D. Let P = {P1, P2,..., Pk} be a partition of point set V (G). For all Pi and Pj in P of order k ≥ 2, i ≠ j, delete the lines between P_i and P_j in G and include the lines between P_i and P_j which are not in G. The resultant graph thus obtained is k−complement of G with respect to the partition P and is denoted by (Formula Presented). For each set V_r in P of order k ≥ 1, delete the lines of G inside V_r and insert the lines of G joining the points of V_r. The graph (Formula Presented) thus obtained is called the k(i)−complement of G with respect to the partition P. In this paper, we define D−complement and D(i)−complement of a graph G. Further we study various properties of D and D(i) complements of a given graph.