Compound Logics for Modification Problems

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence , defining some property of the modified part of the graph, and the target sentence , defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic ( CMSO ) and have models of bounded treewidth, while target sentences express first-order logic ( FO ) properties. Our logic captures problems that are not definable in FO and, moreover, may have instances of unbounded treewidth. Our main result is that, for this compound logic, model-checking can be done in quadratic time on minor-free graphs. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique . It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle’s theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle’s theorem and Gaifman’s theorem. Our algorithmic meta-theorem encompasses, unifies, and extends the known meta-algorithmic results for CMSO and FO on minor-closed graph classes.