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Forbidding induced even cycles in a graph: typical structure and counting

We determine, for all $k\geq 6$, the typical structure of graphs that do not contain an induced $2k$-cycle. This verifies a conjecture of Balogh and Butterfield. Surprisingly, the typical structure of such graphs is richer than that encountered in related results. The approach we take also yields an approximate result on the typical structure of graphs without an induced $8$-cycle or without an induced $10$-cycle.