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In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and G P  k  , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(G P  k  ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(G P  k  ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which G P  k  and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which G P  k  is a graph of rank r and A(G)A(G P  k  ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph K n1,n2,...,nk  which has a commuting decomposition into a perfect matching and its k-complement.

Commuting decomposition of K n1,n2,...,nk through realization of the product A(G)A(G P k )