  We introduce a model describing the paths that pin an elastic interface
moving in a disordered medium. We find that the scaling properties of these
``elastic pinning paths'' (EPP) are different from paths embedded on a directed
percolation cluster, which are known to pin the interface of the ``directed
percolation depinning'' class of surface growth models. The EPP are
characterized by a roughness exponent $\alpha=1.25$, intermediate between that
of the free inertial process ($\alpha=3/2$) and the diode-resistor problem on a
Cayley tree ($\alpha=1$). We also calculate numerically the mean cluster size
and the cluster size distribution for the EPP.
