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Planar Tur\'an number, denoted by ex (n, H), is the maximum number of edges in an n-vertex planar graph which does not contain H as a subgraph. Ghosh, Gyori, Paulos and Xiao initiated the topic of the planar Tur\'an number for double stars. For balanced double star, S₃, ₃ is the only remaining graph need to be considered. In this paper, we give the exact value of ex (n, S₃, ₃), forcing the planar Tur\'an number for all balanced double stars completely determined.
Xu et al. (Sun,) studied this question.