We study the propagation of electromagnetic waves in infinite lines with grafted finite lines in a frequency regime, where the grafted lines have negative phase velocity. The physical properties of such a structure with negative phase velocity materials have not been investigated before, to the best of our knowledge. The theory uses a Green's function formalism. For a single resonator structure we calculate the transmission intensity, and for periodic systems, we determine the transmission intensities and the miniband structures as functions of the number of resonators. In a single resonator structure, resonances are manifested as dips in the transmission, which evolve into stopping minigaps for periodic systems. The dispersion relations of collective normal modes exhibit new allowed minibands or stopping minigaps for the electromagnetic wave propagation, which depend on the number of resonators and the resonator size. These results are in agreement with the features of the transmission spectra. We also investigate the effects of dissipation on the transmission spectra of such structures.