This study numerically investigates the effects of the Weber number (We) and cylinder-to-droplet radius ratio (R*) on the impact dynamics of a low-viscosity droplet on a hydrophobic cylinder by the lattice Boltzmann method. The intrinsic contact angle of the surface is chosen as θ0 = 122°± 2°, which ensures a representative hydrophobicity. The regime diagram of the impact dynamics in the parameter space of We versus R* is established with categories of split and nonsplit regimes. The droplet would split during impact as α = We/R* exceeds a critical value. In the nonsplit regime, the droplet bounces off the cylinder at most Weber numbers unless the impact velocity is minuscule (We < 2). The contact time of the droplet on the cylinder surface decreases with increasing R* or decreasing We, indicating bouncing is facilitated under such conditions. This can be explained by the suppressed adhesion dissipation between the droplet and surface due to a reduction in the contact area. In the split regime, sufficient kinetic energy inside the impacting droplet determines whether the whole droplet could detach from the surface. With a small cylinder (R* < 0.83) and large We (>25), the adhesion effect is weakened for the side fragments because of the small contact area, and it facilitates the dripping of fragments. For other conditions, the detachment, especially for the tiny droplet on the cylinder top, only occurs if the deformation is prominent at We > 35. Moreover, the spreading dynamics of the impacting droplet are also highlighted in this work.