There are two particular Θ₆-graphs - the 6-cycle graphs with a diagonal. We find the planar Tur\'an number of each of them, i.e. the maximum number of edges in a planar graph G of n vertices not containing the given Θ₆ 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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Guan et al. (2024) studied this question.
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