Exact planar Turán number of the double star S_{3,5}

Determine the exact value of the planar Turán number $ex_P(n,S_{3,5})$, the maximum number of edges in an $n$-vertex planar graph containing no copy of the double star $S_{3,5}$.

Background

The paper studies planar Turán numbers for double stars, where the double star $S_{k,l}$ is formed from an edge whose two endpoints are joined to $k$ and $l$ additional leaf vertices, respectively. Prior work established exact values or upper bounds for several smaller double stars, but the exact planar Turán number for $S_{3,5}$ was not known. The paper proves an upper bound of $ex_P(n,S_{3,5})\leq 23n/8-3$ for $S_{3,5}$-free planar graphs that contain no 6-6 edges; it does not determine the exact value of $ex_P(n,S_{3,5})$.