deciding whether a triangle-free graph embedded in a fixed surface (other than the sphere) is $3$-colorable was open. The question was resolved for the projective plane by the result of Gimbel and Thomassen [@gimbel], and in a far reaching generalization, Dvořák, Král’ and Thomas [@trfree7] proved that there exists a linear-time algorithm for this problem for any fixed surface, even if a bounded number of vertices have prescribed colors. In order to design their algorithm, they show in [@trfree6] that every triangle-free graph embedded in a fixed surface exhibits a special structure that determines its $3$-coloring properties. Essentially, they show the following. Let $G$ be a graph embedded in a surface $\Sigma$ with $k$ precolored vertices and without non-facial contractible $4$-cycles. Then $\Sigma$ can be cut in to a bounded number of pieces along curves tracing closed walks in $G$, so that

-   the number of vertices contained in the boundaries of the pieces is bounded by a constant depending only on $\Sigma$ and $k$,

-   every precolored vertex is contained in the boundary of at least one of the pieces, and

-   each piece $\Pi$ and the subgraph $H$ of $G$ drawn in $\Pi$ satisfies one of the following:

    