Courcelle's theorem states that every graph property definable in monadic second-order logic can be decided in linear time on graphs of bounded treewidth. This is one of the most well-known algorithmic meta-theorems.

Motivated by Courcelle's theorem, I made the following conjecture :

Conjecture : Let $\psi$ be any MSO-definable property. If $\psi$ is solvable in polynomial-time on planar graphs, then $\psi$ is solvable in polynomial-time on all classes of minor-free graphs.

I want to know if the above conjecture is obviously false i.e., is there an MSO-definable property that is polynomial-time solvable on planar graphs but NP-hard on some class of minor-free graphs ?

This is the motivation behind my earlier question : Are there any problems that are polynomially solvable on graphs of genus g but NP-hard on graphs of genus > g.


1 Answer 1


Being 4-colorable? Certainly MSO, and trivial on planar graphs. It's NP-complete for a large enough forbidden clique minor, by reduction to planar 3-colorability.

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    $\begingroup$ More explicitly, 4-colorability is NP-complete on the minor-closed family of apex graphs, by reduction to planar 3-colorability. $\endgroup$ Aug 29, 2014 at 20:25

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