N.H. Rev. Stat. Ann. § 265:148

Upon petition of 5 or more legal voters, the selectmen of a town or the mayor of a city, upon notice and hearing, shall have the power to include within the foregoing provisions, sidewalks outside the compact part of the town or city that are built or improved by said town or city or by the abutters; but in such case, notices to that effect shall be posted near said walks at least one week before they shall be so included.