1.  every $3$-coloring of the vertices of $H$ contained in the boundary of $\Pi$ extends to a $3$-coloring of $H$; or,

    2.  a $3$-coloring of the vertices of $H$ contained in the boundary of $\Pi$ extends to a $3$-coloring of $H$ if and only if it satisfies a specific condition (“the winding number constraint”); or,

    3.  $\Pi$ is homeomorphic to the cylinder and all faces of $H$ have length $4$; or,

    4.  $\Pi$ is homeomorphic to the cylinder and both boundary cycles of $H$ have length $4$.

Here, the *cylinder* is the sphere with two holes.

Thus, to determine whether the precoloring extends to a $3$-coloring $G$, we try all the (constantly many) extensions to $3$-colorings of the boundary vertices of the pieces, and then test whether one of them extends to all the pieces. In the first two possibilities for the pieces of the structure we have a complete information about which colorings of the boundaries of the pieces extend, and the last part is thus trivial. However, in the last two cases the information is much more limited. While this is sufficient for the purposes of the algorithm of [@trfree7], it would be preferable to have a more 