Owners of the majority of the frontage on any county road or portion thereof may petition the county legislative authority to vacate and abandon the same or any portion thereof. The petition must show the land owned by each petitioner and set forth that such county road is useless as part of the county road system and that the public will be benefited by its vacation and abandonment. The legislative authority may (1) require the petitioners to make an appropriate cash deposit or furnish an appropriate bond against which all costs and expenses incurred in the examination, report, and proceedings pertaining to the petition shall be charged; or (2) by ordinance or resolution require the petitioners to pay a fee adequate to cover such costs and expenses.