Suppose that $h$ is an instance of the Gaussian free field (GFF) on a simply connected domain $D \subseteq {\mathbf C}$ and $x,y \in \partial D$ are distinct. Fix $κ\in (0,4)$ and for each $θ\in {\mathbf R}$ let $η_θ$ be the flow line of $h$ from $x$ to $y$. Recall that for $θ_1 < θ_2$ the fan ${\mathbf F}(θ_1,θ_2)$ of flow lines of $h$ from $x$ to $y$ is the closure of the union of $η_θ$ as $θ$ varies in any fixed countable dense subset of $[θ_1,θ_2]$. We show that the adjacency graph of components of $D \setminus {\mathbf F}(θ_1,θ_2)$ is a.s. connected, meaning it a.s. holds that for every pair $U,V$ of components there exist components $U_1,\ldots,U_n$ so that $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We further show that ${\mathbf F}(θ_1,θ_2)$ a.s. determines the flow lines used in its construction. That is, for each $θ\in [θ_1,θ_2]$ we prove that $η_θ$ is a.s. determined by ${\mathbf F}(θ_1,θ_2)$ as a set.