When any number of landowners owning land adjacent to any fencing district organized under and pursuant to the law shall present to the county court a petition, in writing, accompanied by a map, giving a description and setting forth what land they desire to have enclosed in such district, and where the lands as set forth and described in the petition are bounded or completely enclosed by existing fencing districts, no-fence districts, counties where a stock law is in effect, or navigable rivers, or combination of them, it shall be the duty of the court to give notice, by publication in some newspaper published in the county where the lands lie, of the filing of the petition and a description of all lands as set forth and contained in the description.