Generalized complements of a graph

Let G = (V, E) be a graph and P = {V₁, V₂, ..., V_k} be a partition of V of order k ≥ 1. For each set V_r in P, remove the edges of G inside V_r and add the edges Ḡ, (the complement of G) joining the vertices V_r. The graph G^P_k (i) thus obtained is called the k(i)-complement of G with respect to P. The graph G is k(i)-self complementary (k(i)-s.c) if G^P_k (i) ≅ G for some partition P of V of order k. Further, G is k(i)co-self complementary (k(i)-co-s.c.) if G^P_k(i) ≅ Ḡ. We determine (1) all k(i)-s.c trees for k = 2, 3, and (2) 2(i)-s.c. unicyclic graphs. Also, some necessary conditions for a tree/unicyclic graph to be k(i)-s.c. are obtained. We indicate how to obtain characterizations of all k(i)-co.s.c. trees, unicyclic graphs and forests from known results.