Dense $2$-connected planar graphs and the planar Turán number of $2C_k$

Abstract: Shi, Walsh and Yu demonstrated that any dense planar graph with certain property (known as circuit graph) contains a large near-triangulation. We extend the result to $2$-connected plane graphs, thereby addressing a question posed by them. Using the result, we prove that the planar Tu\'{a}n number of $2C_k$ is $\left[3-\Theta(k<sup>{\log_23})<sup>{-1}\right]n$ when $k\geq 5$.