Determine minimal obstructions for bounded treewidth

Determine all minimal graph-minor obstructions to treewidth at most a fixed integer k, including the case k=4, in order to characterize classes of graphs having bounded parameter \(\p_{a^{k+1}s_{+\infty}}\).

Background

In the conclusion, the paper introduces a broad family of graph parameters $\p_L$, defined from languages over operations corresponding to vertex addition and clique-sums. For the language ak+1s+∞a^{k+1}s_{+\infty}, the resulting parameter equals treewidth plus one on graphs of treewidth at most kk, and is infinite otherwise. Consequently, characterizing graph classes of bounded $\p_{a^{k+1}s_{+\infty}}$ would require identifying the minimal excluded minors for treewidth at most kk.

The paper explicitly notes that these minimal obstructions have not yet been determined, even when k=4k=4. This is presented as a concrete obstruction to obtaining a general characterization of boundedness for the wider family of parameters considered.

References

There is little hope that a general answer can be given to this problem. Indeed, for every nonnegative integer $k$, for every graph $G$,

\p_{a{k+1}s_{+\infty}}(G) = \begin{cases} \tw(G)+1 &\textrm{if $\tw(G) \leq k$} \ +\infty &\textrm{otherwise} \end{cases}

and so a class of graphs has bounded $\p_{a{k+1} s_{+\infty}}$ if and only if it excludes all the minimal obstructions to treewidth at most $k$, which have not been determined yet, even for $k=4$ (see).

— Excluding a rectangular grid  (2501.11617 - Rambaud, 20 Jan 2025) in Section Conclusion