  We develop an ab initio formalism for dipolar electron-phonon interactions
(EPI) in two-dimensional (2D) materials. Unlike purely longitudinal Fr\"ohlich
model, we show that the out-of-plane dipoles also contribute to the
long-wavelength non-analytical behavior of EPI. And the 2D dipolar EPI plays an
important role not only in the typical polar material MoS$_2$, but also in
graphane and fluorinated graphene. By incorporating this formalism into
Wannier-Fourier interpolation, we enable accurate EPI calculations for 2D
materials and subsequent intrinsic carrier mobility prediction. The results
show that Fr\"ohlich model is inadequate for 2D materials and correct
long-wavelength interaction must be included for the reliable prediction.
