Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.

@article{paperbot4034,
  title = {Hardness of monadic second-order formulae over succinct graphs},
  author = {Guilhem Gamard and Aliénor Goubault-Larrecq and Pierre Guillon and Pierre Ohlmann and Kévin Perrot and Guillaume Theyssier},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 22, Issue 1},
  year = {2026},
  doi = {10.46298/lmcs-22(1:3)2026}
}