The structure and the number of $P_7$-free bipartite graphs

We show that the number of labelled $P_7$-free bipartite graphs with $n$ vertices grows as $n^{Θ(n)}$. This resolves an open problem posed by Allen [P. Allen, Forbidden induced bipartite graphs. J. Graph Theory 60 (2009), no. 3, 219--241.], and completes the description of speeds of monogenic classes of bipartite graphs. Our solution is based on a new decomposition scheme of bipartite graphs, which is of independent interest.

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