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Paper detail

Hardness of monadic second-order formulae over succinct graphs

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.

preprint2026arXivOpen access
Guilhem GamardAliénor Goubault-LarrecqPierre GuillonPierre OhlmannKévin PerrotGuillaume Theyssier
Logic in Computer ScienceComputational Complexity
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Guilhem GamardAliénor Goubault-LarrecqPierre GuillonPierre OhlmannKévin PerrotGuillaume Theyssier

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