  In this paper, we prove that the path ideals of both paths and cycles have
minimal cellular resolutions. Specifically, these minimal free resolutions
coincide with the Barile-Macchia resolutions for paths, and their generalized
counterparts for cycles. Furthermore, we identify edge ideals of cycles as a
class of ideals that lack a minimal Barile-Macchia resolution, yet have a
minimal generalized Barile-Macchia resolution.
