Ohio Rev. Code § 1905.08
The chief of police of the city or village or a police officer of the city or village designated by him, or the marshal of a village shall attend the sittings of the mayor's court to execute the orders and process of the court, and to preserve order in it. The chief of police, other police officer, or marshal shall execute and return all writs and process directed to him by the mayor. The jurisdiction of the chief of police, other police officer, or marshal in the execution of such writs and process is coextensive with the county in criminal cases and in cases of violations of ordinances of the municipal corporation. In serving such writs and process and taxing costs on them, the chief of police, other police officer, or marshal shall be governed by the laws pertaining to constables. The fees of the mayor are the same as those allowed in the municipal or county court within whose jurisdiction the municipal corporation is located. There shall be allowed and taxed for services of the chief of police, other police officer, or marshal, the same fees and expense as those allowed constables.
R.C. § 1905.08