Planar Turán number of quasi-double stars

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 P ( n , H ) , is the maximum number of edges in a planar H -free graph on n vertices. A ( h, k )-quasi-double star W h,k , obtained from a path P 3 = v 1 v 2 v 3 by adding h leaves and k leaves to the vertices v 1 and v 3 , respectively, is a subclass of caterpillars. In this paper, we study ex P ( n , W h , k ) for all 1 ≤ h ≤ 2 ≤ k ≤ 5, and obtain that ex P ( n , W h , k ) = 3 ( h + k ) h + k + 2 n for 3 ≤ h + k ≤ 5 and ex P ( n , W 1 , 5 ) = 5 2 n . Furthermore, we establish that 9 4 n ≤ ex P ( n , W 2 , 4 ) ≤ 5 2 n and 5 2 n ≤ ex P ( n , W 2 , 5 ) ≤ 17 6 n .