The Graph Minor Structure Theorem through Bidimensionality

The bidimensionality of a set of vertices $X$ in a graph $G$ is the maximum $k$ for which $G$ contains as a $X$-rooted minor the $(k \times k)$-grid. This notion allows for the following version of the Graph Minors Structure Theorem (GMST) that avoids the use of apices and vortices: $K_k$-minor free graphs are those that admit tree decompositions whose torsos contain sets of bounded bidimensionality whose removal yield a graph embeddable in some surface $\Sigma$ of bounded Euler-genus. We next fix the target condition by demanding that $\Sigma$ is some particular surface. This defines a "surface extension" of treewidth, where $\Sigma$-${\sf tw}$ is the minimum $k$ for which $G$ admits a tree decomposition whose torsos become embeddable in $\Sigma$ after the removal of a set of bidimensionality at most $k$. We identify a finite collection $\mathfrak{D}_{\Sigma}$ of parametric graphs and prove that the minor-exclusion of the graphs in $\mathfrak{D}_{\Sigma}$ determines the behavior of $\Sigma$-${\sf tw},$ for every surface $\Sigma.$ It follows that the collection $\mathfrak{D}_{\Sigma}$ bijectively corresponds to the "surface obstructions" for $\Sigma,$ i.e., surfaces that are minimally non-contained in $\Sigma.$ Our results are tight in the sense that $\Sigma$-${\sf tw}$ cannot be bounded for all parametric graphs in $\mathfrak{D}_{\Sigma}$.