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Gromov asked whether every one‐ended word‐hyperbolic group contains a hyperbolic surface group. We prove that every one‐ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank 2 or (2) every generator is used the same number of times in a minimal automorphic image of the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).

Hyperbolic surface subgroups of one‐ended doubles of free groups