Commuting graphs and their generalized complements

In this paper we consider a graph G, a partition P = {V₁, V₂,: :: , V_κ} of V (G) and the generalized complements G^P _κ and G^P _κ(i) with respect to the partition P. We derive conditions to be satisfied by P so that G commutes with its generalized complements. Apart from the general characterization, we also obtain conditions on P = {V₁, V₂,: :: , V_κ} so that G commutes with its generalized complements for certain classes of graphs namely complete graphs, cycles and generalized wheels. In the process we obtain a commuting decomposition of regular complete k-partite graph K_n1,n2,:::,nk in terms of a Hamiltonian cycle and its kcomplement. We also get a commuting decomposition of a complete k-partite graph K_n1,n2,:::,nk in terms of a generalized wheel and its κ- complement, where _n1,n2,:::,nk satisfy some conditions.