Let G a finite group and S be subset of a group G such that 1 S and S = S -1 , the Cayley graph ? = Cay(G, S) of G with respect to S is defined as the graph with vertex set G and edge set ( x, sx) x G, s S} . For a Cayley graph Cay(G, S) of a finite group G , it is called a CI-graph if for any another Cayley graph Cay(G, T) whenever Cay(G, S) Cay(G, T) , there exists an automorphism ? Aut(G) such that S ? = T . Let R , S , T be subsets of a group G such that R = R -1 , S = S -1 and 1 R . Define the undirected graph BCay(G; R , S , T) to have vertex set G 0 , 1} , and with vertices (h , i), (g , j) adjacent if and only if one of the following three possibilities occurs: (1) i = j = 0 and gh -1 R (2) i = j = 1 and gh -1 (3) i = 0 , j = 1 and gh -1 T .