It is the duty of landowners or operators with land adjoining regularly traveled county and township highways, as designated by the township board of supervisors in organized townships, the board of county commissioners in unorganized townships, and the board of county commissioners in the case of county highways, to cut all weeds and grasses along the regularly traveled highways adjoining their lands, including weeds and grasses growing within the public right of way bordering the highways and their lands. The cutting shall be completed not later than September fifteenth or October first, as prescribed by the board of county commissioners. The board of county commissioners may also, if necessary, require an additional cutting be completed by August first.