The width of the crack interface in the fuse model after breakdown, w, scales with the size of the network, L, as w\ensuremath{\sim}${\mathit{L}}^{\mathrm{\ensuremath{\zeta}}}$. When the disorder is narrow, or when it includes arbitrarily small threshold values, we find that \ensuremath{\zeta}=0.7 to within 10%, indicative of this being a universal value. This is not far from 2/3, suggested by an analogy with the random directed polymer problem. When, on the other hand, the disorder is strong and includes arbitrarily large threshold values, the exponent \ensuremath{\zeta} depends on the disorder. These results suggest that the random polymer problem may be relevant for brittle fracture in real materials.