Given a graph H, we call a graph H-free if it does not contain H as a subgraph. The planar Turán number of a graph H, denoted by ex (n, H), is the maximum number of edges in a planar H-free graph on n vertices. A (h, k) -quasi-double star W₇, ₊, obtained from a path P₃=v₁v₂v₃ by adding h leaves and k leaves to the vertices v₁ and v₃, respectively, is a subclass of caterpillars. In this paper, we study ex (n, W₇, ₊) for all 1 h 2 k 5, and obtain some tight bounds ex (n, W₇, ₊) 3 (h+k) h+k+2n for 3 h+k 5 with equality holds if (h+k+2) n, and ex (n, W₁, ₅) 52n with equality holds if 12 n. Also we show that 94n ex (n, W₂, ₄) 52n and 52n ex (n, W₂, ₅) 176n, respectively.
Liu et al. (Wed,) studied this question.