Excluding Surfaces as Minors in Graphs

• Main Authors: Thilikos, Dimitrios M, Wiederrecht, Sebastian
• Format: Journal Article
• Language: English
• Published: 02-06-2023
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2306.01724

We introduce an annotated extension of treewidth that measures the
contribution of a vertex set $X$ to the treewidth of a graph $G.$ This notion
provides a graph distance measure to some graph property $\mathcal{P}$: A
vertex set $X$ is a $k$-treewidth modulator of $G$ to $\mathcal{P}$ if the
treewidth of $X$ in $G$ is at most $k$ and its removal gives a graph in
$\mathcal{P}.$This notion allows for a version of the Graph Minors Structure
Theorem (GMST) that has no need for apices and vortices: $K_k$-minor free
graphs are those that admit tree-decompositions whose torsos have
$c_{k}$-treewidth modulators to some surface of Euler-genus $c_{k}.$ This
reveals that minor-exclusion is essentially tree-decomposability to a
``modulator-target scheme'' where the modulator is measured by its treewidth
and the target is surface embeddability. We then fix the target condition by
demanding that $\Sigma$ is some particular surface and define a ``surface
extension'' of treewidth, where $\Sigma\mbox{-}\mathsf{tw}(G)$ is the minimum
$k$ for which $G$ admits a tree-decomposition whose torsos have a $k$-treewidth
modulator to being embeddable in $\Sigma.$We identify a finite collection
$\mathfrak{D}_{\Sigma}$ of parametric graphs and prove that the minor-exclusion
of the graphs in $\mathfrak{D}_{\Sigma}$ precisely determines the asymptotic
behavior of ${\Sigma}\mbox{-}\mathsf{tw},$ 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 $\Sigma.$