Ohio Rev. Code Ann. § 5555.82

The proportion of the cost and expense of a road improvement on or along the line between this state and an adjoining state to be paid by any county in this state may be assessed or levied upon the property of this state by the board of county commissioners of such county in any of the ways provided in sections 5555.01 to 5555.72 of the Revised Code, in the case of an improvement wholly within this state. Such board may also join with the department of transportation of this or other states in the construction of such road as if the improvement were wholly within this state. Such board may also join with the department of transportation of this or other states in the construction of such road as if the improvement were wholly within this state.