The planar Tur\'an number of $\{K_{4},\Theta_{6}^{i}\}$
Let $\mathcal{H}$ be a family of graphs. A graph is said to be $\mathcal{H}$-free if it contains no subgraph isomorphic to a graph in $\mathcal{H}$. The planar Tur\'an number $ex_{_\mathcal{P}}(n,\mathcal{H})$ is defined as the maximum number of edges in an $\mathcal{H}$-free planar graph on $n$ vertices. In this paper, we determine the exact value of $ex_{_\mathcal{P}}(n,\{K_{4}, \Theta_{6}^{1}\})$ and a tight upper bound of $ex_{_\mathcal{P}}(n,\{K_{4}, \Theta_{6}^{2}\})$.
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