Characterization of generalized complements of a graph

For a graph G(V, E), let P = {V₁, V₂, V₃, …, V_k } be a partition of vertex set V (G) of order k ≥ 2. For all V_i and V_j in P, i ≠ j, remove the edges between V_i and V_j in graph G and add the edges between V_i and V_j which are not in G. The graph G^P_k thus obtained is called the k−complement of graph G with respect to the partition P. For each set V_r in P, remove the edges of graph G inside V_r and add the edges of G (the complement of G) joining the vertices of V_r. The graph G^P_k(i) thus obtained is called the k(i)−complement of graph G with respect to the partition P. In this paper, we characterize few properties of generalized complements of a graph.