Universal excluded-minor obstructions for bounded language-defined graph parameters

Characterize classes of graphs having bounded fp_Lf in terms of excluded minors for every language fL \subseteq \Sigma^*f, where f\Sigma = \{a\} \cup \{s_k \mid k \ge 1\}f and fp_Lf is the graph parameter defined by minimizing the vertex-addition complexity over words in fLf.

Background

The paper defines a broad family of minor-monotone graph parameters fp_Lf from languages over an alphabet whose symbol faf represents vertex addition and whose symbol fs_kf represents clique-sums along cliques of size less than fkf. This framework includes treewidth, fkf-treedepth, vertex-cover number, feedback vertex set number, and related parameters.

The proposed problem asks for an excluded-minor characterization of boundedness for every such language-defined parameter. The paper notes that a general solution is difficult: even the relevant minimal obstructions for treewidth bounded by a fixed constant have not been determined in general, with the case of treewidth at most four cited as unresolved. The authors further explain that the problem is connected to the search for universal obstructions for broad families of graph parameters.