  The roughness properties of two-dimensional fracture surfaces as created by
the slow failure of random fuse networks are considered and compared to yield
surfaces of perfect plasticity with similar disorder. By studying systems up to
a linear size L=350 it is found that in the cases studied the fracture surfaces
exhibit self-affine scaling with a roughness exponent close to 2/3, which is
asymptotically exactly true for plasticity though finite-size effects are
evident for both. The overlap of yield or minimum energy and fracture surfaces
with exactly the same disorder configuration is shown to be a decreasing
function of the system size and to be of a rather large magnitude for all cases
studied. The typical ``overlap cluster'' length between pairs of such
interfaces converges to a constant with $L$ increasing.
