Hardness of monadic second-order formulae over succinct graphs

Guilhem Gamard; Aliénor Goubault-Larrecq; Pierre Guillon; Pierre Ohlmann; Kévin Perrot; Guillaume Theyssier

doi:10.46298/lmcs-22(1:3)2026

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arXiv · 2302.04522

Hardness of monadic second-order formulae over succinct graphs

Guilhem Gamard·Aliénor Goubault-Larrecq·Pierre Guillon·Pierre Ohlmann·Kévin Perrot·Guillaume Theyssier

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Abstract

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.

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Guilhem Gamard, Aliénor Goubault-Larrecq, Pierre Guillon, Pierre Ohlmann, Kévin Perrot, Guillaume Theyssier. 2023-02-09. Hardness of monadic second-order formulae over succinct graphs. https://doi.org/10.46298/lmcs-22(1%3A3)2026

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