The planar Turán number of $\{K_4,{\mathrm{\Theta }}_5\}$

Let $\mathcal{F}$ be a set of graphs. The planar Turán number, $e{x}_{\mathcal{P}}(n,\mathcal{F})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{F}$ as a subgraph. In this paper, we give upper bounds of $e{x}_{\mathcal{P}}(n,\{K_4,{\mathrm{\Theta }}_5\})⩽25\mathrm{/}11(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.