An intriguing conjecture of Gromov asserts that every one-ended word-hyperbolic group G contains a surface group. A simple but still influential case is when G splits as the double of a free group amalgamated along a cyclic subgroup. In the first part of the talk, I will describe a combinatorial property (called, polygonality) for a word in a free group and show that polygonality guarantees that the corresponding double contains a surface group. Proving polygonality is much like a jigsaw puzzle in free groups, and conjecturally, this can resolve Gromov conjecture for many graphs of free groups with cyclic edge groups. In the second part, I will exhibit surprisingly many words that are polygonal, including words that can be realized as simple curves on the boundary of a handlebody. Using Whitehead graphs, I will formulate a graph theoretic formulation of polygonality and describe an algorithm that can decide polygonality in finite time. The first part is joint work with Henry Wilton.