Slowing down DNA translocation speed in a nanopore is essential to ensuring reliable resolution of individual bases. Thin membrane materials enhance spatial resolution but simultaneously reduce the temporal resolution as the molecules translocate far too quickly. In this study, the effect of exposed graphene layers on the transport dynamics of both single (ssDNA) and double-stranded DNA (dsDNA) through nanopores is examined. Nanopore devices with various combinations of graphene and Al2O3 dielectric layers in stacked membrane structures are fabricated. Slow translocations of ssDNA in nanopores drilled in membranes with layers of graphene are reported. The increased hydrophobic interactions between the ssDNA and the graphene layers could explain this phenomenon. Further confirmation of the hydrophobic origins of these interactions is obtained through reporting significantly faster translocations of dsDNA through these graphene layered membranes. Molecular dynamics simulations confirm the preferential interactions of DNA with the graphene layers as compared to the dielectric layer verifying the experimental findings. Based on our findings, we propose that the integration of multiple stacked graphene layers could slow down DNA enough to enable the identification of nucleobases.