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Graphene wires have been fabricated from large-area multilayer graphene sheets grown by chemical vapor deposition. As the methane concentration increases, a larger percentage of thicker graphene layers are grown. The multilayer graphene sheets have an average thickness of 10-20 nm with sheet resistances between 500 and 1000 Ω/sq. The sheet resistance shows a strong correlation with the average surface roughness. This letter reports measured breakdown current densities up to 4×107 A/cm2, where resistive heating is proposed as the main breakdown mechanism. Increasing the uniformity of the graphene layers is important in achieving a higher breakdown current density.