Ohio Rev. Code § 5557.07
Whenever any portion of a road to be improved by the board of county commissioners lies within the corporate limits of a municipal corporation, and the legislative authority of such municipal corporation desires to improve any part of the road within such municipal corporation to a greater width than is contemplated by the proceedings for the improvement by the board, such legislative authority shall, at any time before it approves the surveys and profiles for such improvement, determine by resolution the additional width to which it desires such road, or part thereof, to be improved and cause a copy of the resolution to be filed with the board. The board shall cause the county engineer to prepare the necessary surveys, plans, profiles, cross sections, estimates, and specifications for improving such road or part thereof to an additional width. The estimates for the improvement shall set forth the probable cost of so much of the improvement as is made necessary by the proposed increase in width. After such surveys, plans, profiles, cross sections, estimates, and specifications have been completed by the engineer he shall file a copy with the board and all subsequent proceedings shall be conducted in the manner provided in sections 5557.02 to 5557.06, inclusive, of the Revised Code. The legislative authority of the municipal corporation may assume all that part of the estimated cost of such improvement rendered necessary by the increased width, or such part of the additional expense as may be agreed upon between the legislative authority and the board.
R.C. §5557.07