Planar Turán numbers of three configurations

Abstract: The planar Tu\'{a}n number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for example, $H$ is a long path or a long cycle), while tight bounds have been established for many small planar graphs such as cycles, paths, $\Theta$-graphs and other small graphs formed by a union of them. One representative graph among such union graphs is $K_1+L$ where $L$ is a linear forest without isolated vertices. Previous works solved the cases when $L$ is a path or a matching. In this work, we first investigate the planar Tur\'{a}n number of the graph $K_1+L$ when $L$ is the disjoint union of a $P_2$ and $P_3$. Equivalently, $K_1+L$ represents a specific configuration formed by combining a $C_3$ and a $\Theta_4$. We further consider the planar Tur\'{a}n numbers of the all graphs obtained by combining $C_3$ and $\Theta_4$. Among the six possible such configurations, three have been resolved in earlier works. For the remaining three configurations (including $K_1+(P_2\dot{\cup}P_3)$), we derive tight bounds. Furthermore, we completely characterize all extremal graphs for the remaining two of these three cases. Additionally, for two other unsolved cases involving $K_1+L$, we establish improved bounds.