There are two particular $Θ_6$-graphs - the 6-cycle graphs with a diagonal. We find the planar Turán number of each of them, i.e. the maximum number of edges in a planar graph $G$ of $n$ vertices not containing the given $Θ_6$ as a subgraph and we find infinitely many extremal constructions showing the sharpness of these results - apart from a small additive constant error in one of the cases.
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