CARBON IS PRESENT AS GRAPHITE IN WHITE CAST IRON

you are right, carbon is present as graphite in white cast iron for inthe time of extraction  and acts as cathode

Cast iron is iron or a ferrous alloy which has been heated until it liquefies, and is then poured into a mould to solidify. It is usually made from pig iron. The alloy constituents affect its colour when fractured: white cast iron has carbide impurities which allow cracks to pass straight through. Grey cast iron has graphitic flakes which deflect a passing crack and initiate countless new cracks as the material breaks.

carbon is present as cementite(Fe3C) in white cast iron.............where as in grey cast iron it is present as Graphite........hope you understand......