No owner of real estate within the corporate limits of such city shall be permitted to subdivide the real estate into blocks and lots, or parcels, without first having obtained from the city engineer a plat or plan for the avenues, streets, and alleys to be laid out within or across the same and, when applicable, having complied with sections 39-1311 to 39-1311.05. A copy of such plat must be filed in the office of the city clerk for at least two weeks before such plat can be approved. Public notice must be given for two weeks of the filing of the plat, and such plat, if ordered by the council, shall be made so that such avenues, streets, and alleys so far as practicable, shall correspond in width, name, and direction and be continuous of the avenues, streets, and alleys in the city contiguous to or near the real estate to be subdivided. The council shall have power to compel the owner of such real estate, in subdividing the same, to lay out and dedicate to the public the avenues, streets, and alleys, to be within or across such real estate in accordance with the plat. It shall further have the power to prohibit the selling or offering for sale of any lots or parts of such real estate not subdivided and platted as herein required. It shall also have power to establish the grade of all such streets and alleys and to require the same to be graded to such established grade before selling or offering for sale any of the lots or parts of the real estate. Any and all additions to be made to the city shall be made so far as the same relates to the avenues, streets, and alleys therein, under and in accordance with the foregoing provisions. Whenever the owners of all the lots and lands, except streets and alleys, embraced and included in any existing plat or subdivision shall desire to vacate the plat or subdivision for the purpose of replatting the land embraced in the plat or subdivision, and shall present a petition praying for such vacation to the city council, and submit therewith for the approval of the city council a proposed replat of the same, which shall in all things be in conformity with the requirements of this section, the city council may, by concurrent resolution, declare the existing plat and the streets and alleys therein vacated and approve the proposed replat. Thereupon the existing plat or subdivision shall be vacated and the land comprised within the streets and alleys so vacated shall revert to and the title thereto vest in the owners of the abutting property and become a part of such property, each owner taking title to the centerline of the vacated street or alley adjacent to his or her property. When a portion of a street or alley is vacated only on one side of the center thereof, the title to such land shall vest in the owner of the abutting property and become a part of such property. It shall require a two-thirds vote of all the members of the city council to adopt such resolution. Upon the vacation of any plat as aforesaid, it shall be the duty of the owners petitioning for same to cause to be recorded in the office of the register of deeds and county assessor of the county a duly certified copy of the petition, the action of the council therein, and the resolution vacating the plat.