7 problems

Cliquewidth preservation conjecture. There is a function $f$ such that every graph in $\mathcal C$ has cliquewidth at most $f(a)$.

General decomposition conjecture. The theorem can be broadly generalized to a general decomposition result for all dependent unstable classes of graphs.

Clique-width characterization conjecture. The class $\mathscr C$ has bounded cliquewidth if and only if it has a modelization in a class of colored posets whose cover graphs have b…

Linear cliquewidth CMSO obstruction conjecture. A class of graphs $\mathscr{C}$ has bounded linear cliquewidth if and only if the class of trees cannot be $\mathsf{CMSO}$-transduce…

Gajarsky–Pilipczuk–Toruńczyk's cliquewidth obstruction conjecture. A class of graphs $\mathscr{C}$ has unbounded cliquewidth if and only if $\mathscr{C}$ transduces a class…

Let $\mathscr{C}$ be a class of graphs, and consider the satisfiability problem for monadic second-order logic ($\mathsf{MSO}$) on $\mathscr{C}$. Seese's bounded-cliquewidth conjec…

Let $S'$ be a finite set of graphs, and let $\operatorname{Forb}_\subseteq(S')$ be the family of graphs containing no member of $S'$ as an induced subgraph. Wqo-bounded-cliquewidth…