Tropical cyclones (TC) are often generated from pre-existing "seed" vortices. Seeds with higher persistence might have a higher chance to undergo TC genesis. What controls seed persistence remains unclear. This study proposes that planetary Rossby wave drag is a key factor that affects the persistence of the seed. Using recently developed theory for the response of a vortex to the planetary vorticity gradient (β), a new parameter given by the ratio of the maximum wind speed to the Rhines speed at the radius of maximum wind, here termed "vortex compactness" (C_v), is introduced to characterize the vortex weakening by planetary Rossby wave drag. The relationship between vortex compactness and weakening rate is tested using barotropic β-plane experiments. Results from these barotropic experiments suggest that there is a robust relationship between the initial compactness of the vortex and the weakening rate due to planetary Rossby wave drag. Next, an analysis of TC seeds in the reanalysis data reveals a significant difference in the distribution of lifetime-averaged C_v between developing and non-developing seeds. The fact that most developing seeds are compact and most incompact seeds are non-developing results in a positive relationship between the lifetime-averaged C_v of TC seeds and their TC genesis potential. The maximum persistence of incompact seeds is successfully predicted by their lifetime-averaged C_v, and exhibits the same relationship found in barotropic experiments. These results suggest that a seed's structure strongly modulates how long it can persist in the presence of a planetary vorticity gradient. The analysis is further extended using a climate model hierarchy to further elucidate the impact of planetary Rossby wave drag on the persistence of TC seeds, and to underscore the significance of seed structure in their genesis potential.