Some results on generalized self-complementary graphs

Let P = {V1,V2,...,Vk} be a partition of vertex set V of order k ≥ 2 of a graph G(V,E). The k-complement of G denoted by GkP is defined as for all Vi and Vj in P, i ≠ j, remove the edges between Vi and Vj in G and add the edges between Vi and Vj which are not in G. The graph G is called k-self-complementary if G≅GkP. For a graph G, k(i)-complement of G denoted by Gk(i)P is defined as for each Vr remove the edges of G inside Vr and add the edges of G¯ by joining the vertices of Vr. The graph G is called k(i)-self-complementary if G≅Gk(i)P for some partition P of order k. In this paper, we determine generalized self-complementary graphs of forest, double star and unicyclic graphs.