  We give an exact analytical results for diffusion with a power-law position
dependent diffusion coefficient along the main channel (backbone) on a comb and
grid comb structures. For the mean square displacement along the backbone of
the comb we obtain behavior $\langle x^2(t)\rangle\sim t^{1/(2-\alpha)}$, where
$\alpha$ is the power-law exponent of the position dependent diffusion
coefficient $D(x)\sim |x|^{\alpha}$. Depending on the value of $\alpha$ we
observe different regimes, from anomalous subdiffusion, superdiffusion, and
hyperdiffusion. For the case of the fractal grid we observe the mean square
displacement, which depends on the fractal dimension of the structure of the
backbones, i.e., $\langle x^2(t)\rangle\sim t^{(1+\nu)/(2-\alpha)}$, where
$0<\nu<1$ is the fractal dimension of the backbones structure. The reduced
probability distribution functions for both cases are obtained by help of the
Fox $H$-functions.
