Decidability in geometric grid classes of permutations

We prove that the basis and the generating function of a geometric grid class of permutations Geom$(M)$ are computable from the matrix $M$, as well as some variations on this result. Our main tool is monadic second-order logic on permutations and words.Comment: 14 page

Graph Parameters, Universal Obstructions, and WQO

We introduce the notion of universal obstruction of a graph parameter, with respect to some quasi-ordering relation. Universal obstructions may serve as compact characterizations of the asymptotic behavior of graph parameters. We provide order-theoretic conditions which imply that such a characterization is finite and, when this is the case, we present some algorithmic implications on the existence of fixed-parameter algorithms

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 (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or $\mathcal{G}$-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. 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. We finally prove extensions where the target property is expressible in FOL+DP, i.e., the enhancement of FOL with disjoint-paths predicates

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum