The Graph Minors Structure Theorem through Bidimensionality

Abstract

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 some $(k \times k)$-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 $Σ$ of bounded Euler-genus. We next fix the target condition by demanding that $Σ$ is some particular surface. This defines a "surface extension" of treewidth, where $Σ\mbox{-}\textsf{tw}(G)$ is the minimum $k$ for which $G$ admits a tree-decomposition whose torsos become embeddable embeddable in $Σ$ after the removal of a set of dimensionality at most $k$. We identify a finite collection $\mathfrak{D}_Σ$ of parametric graphs and prove that the minor-exclusion of the graphs in $\mathfrak{D}_Σ$ determines the behavior of $Σ\mbox{-}\textsf{tw}$, for every surface $Σ.$ It follows that the collection $\mathfrak{D}_Σ$ bijectively corresponds to the "surface obstructions" for $Σ,$ i.e., surfaces that are minimally non-contained in $Σ.$ Our results are tight in the sense that $Σ\mbox{-}\textsf{tw}$ cannot be bounded for all parametric graphs in $\mathfrak{D}_Σ$.

Figures (12)

• Figure 1: The annulus grid $\mathscr{A}_{9},$ the handle grid $\mathscr{H}_{9}$ and the crosscap grid $\mathscr{C}_{9}$ in order from left to right. Notice that both $\mathscr{H}_{9}$ and $\mathscr{C}_{9}$ contain two $(18\times 9)$ grids as vertex-disjoint subgraphs (depicted in different colors). Moreover, $\mathscr{C}_{9}$ contains a $(18\times 18)$-grid as a spanning subgraph.
• Figure 2: The Dyck-grid of order $8$ with one handle and two crosscaps, i.e., the graph $\mathscr{D}_{8}^{1,2}.$
• Figure 3: The lattice of surface containment, along with some indicative closed sets of surfaces $\mathbb{S}$ (in grey) and their surface obstructions $\mathsf{sobs}(\mathbb{S})$ (depicted in bold frames). Notice that, in case (c), $\mathbb{S}$ is the set of surfaces contained in the Klein bottle $\Sigma^{(0,2)}$ and its the surface obstruction set contains only the torus, i.e., $\mathsf{sobs}(\mathbb{S})=\{\Sigma^{(1,0)}\}.$ For each set of surfaces, the orange box indicates the corresponding prevalent surface.
• Figure 4: A rooted $5$-wall $(\widehat{W},X).$ The edges and the internal nodes of its hairs are orangle. The set $S$ is depicted by the magenta vertices and the set $X$ is depicted by the blue circles. The branch vertices are the square vertices.
• Figure 5: The graph $\widetilde{\mathscr{D}}_{8}^{1,2}$ and a half-integral packing of four subdivisions of $\widetilde{\mathscr{D}}_{2}^{1,2},$ colored by different colors.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Proposition 1
• Theorem 1.3
• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Theorem 1.6
• Theorem 1.7