Some properties of generalized complements of a graph

Let P = {V₁,V₂, · · ·,V_k } be a partition of vertex set V of G. The k−complement of G denoted by G^Pk^is defined as follows: for all V_i and V_j in P, i ≠ j, remove the edges between V_i and V_j and add edges between V_i and V_j which are not in G. The graph G is k-self complementary with respect to P if G^P∼k = G. The k(i)-complement G^Pk(i)of a graph G with respect to P is defined as follows: for all Vr ∈ P, remove edges inside V_r and add edges which are not in V_r. In this paper we provide sufficient conditions for G^Pk and G^Pk(i) to be disconnected, regular, line preserving and Eulerian.