The Department of Highways is hereby authorized to enter into agreements with cities of the second class, second class A, and third class providing for the improvement, construction, reconstruction and/or maintenance, in whole or in part, of the existing width, or of a lesser width, of any streets and highways in any such city which are not on the plan of the State highway system but which are continuations of State highways entering such cities, or running through such cities, or which furnish the shortest or most convenient route through such cities for the traveling public, and such agreements may provide that the improvement, construction, reconstruction and/or maintenance shall be done by the Department of Highways, or the city, or by contract let by the Department of Highways, or by the city, or by both, and that the Commonwealth shall, in either event, pay the whole or any portion of the cost of such improvement, construction, reconstruction and/or maintenance, the city to pay the remaining portion of such cost. Any city of the second class, second class A, or third class is hereby authorized to enter into any contract, as hereinbefore provided for, with the Department of Highways, and to expend the moneys of the city for such purposes. Any such agreement may also provide for the improvement, construction or reconstruction of a greater width of any such street or highway than that agreed to be paid for wholly or partly by the Commonwealth, and for the payment by the city, or by the assessment of benefits as hereinafter provided, or both, of the whole cost of any such additional width.