  In the so-called (2,2) theory, which is the U(N)^4 circular quiver
superconformal Chern-Simons theory with levels (k,0,-k,0), it was known that
the instanton effects are described by the free energy of topological strings
whose Gopakumar-Vafa invariants coincide with those of the local D_5 del Pezzo
geometry. By considering two types of one-parameter rank deformations U(N) x
U(N+M) x U(N+2M) x U(N+M) and U(N+M) x U(N) x U(N+M) x U(N), we classify the
known diagonal BPS indices by degrees. Together with other two types of
one-parameter deformations, we further propose the topological string
expression when both of the above two deformations are turned on.
