  L\'evy-type walks with correlated jumps, induced by the topology of the
medium, are studied on a class of one-dimensional deterministic graphs built
from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a
standard random walk on the sets but is also allowed to move ballistically
throughout the empty regions. Using scaling relations and the mapping onto the
electric network problem, we obtain the exact values of the scaling exponents
for the asymptotic return probability, the resistivity and the mean square
displacement as a function of the topological parameters of the sets.
Interestingly, the systems undergoes a transition from superdiffusive to
diffusive behavior as a function of the filling of the fractal. The
deterministic topology also allows us to discuss the importance of the choice
of the initial condition. In particular, we demonstrate that local and average
measurements can display different asymptotic behavior. The analytic results
are compared with the numerical solution of the master equation of the process.
