🤖 AI Summary
This paper investigates the logical characterization of graph class complexity, addressing the central question: can first-order (FO) transductions replace monadic second-order (MSO) transductions as the foundational tool for structural comparison and classification of graph classes? To this end, it introduces a local paradigm for FO transductions and systematically studies the quasiorder induced by FO interpretability. Key contributions include: (i) the first proof that the pathwidth hierarchy is strictly increasing under FO transductions; (ii) the demonstration that this quasiorder is non-lattice and lacks minimal or maximal graph classes; (iii) the establishment of sparse graph classes—particularly those of bounded treewidth—as pivotal anchors in the FO transduction quasiorder; and (iv) a complete characterization of FO transduction relationships among fundamental graph classes, including forests, planar graphs, and interval graphs. These results provide a finer-grained, model-theoretically richer logical framework for graph complexity theory.
📝 Abstract
We study various aspects of the first-order transduction quasi-order on graph classes, which provides a way of measuring the relative complexity of graph classes based on whether one can encode the other using a formula of first-order (FO) logic. In contrast with the conjectured simplicity of the transduction quasi-order for monadic second-order logic, the FO-transduction quasi-order is very complex, and many standard properties from structural graph theory and model theory naturally appear in it. We prove a local normal form for transductions among other general results and constructions, which we illustrate via several examples and via the characterizations of the transductions of some simple classes. We then turn to various aspects of the quasi-order, including the (non-)existence of minimum and maximum classes for certain properties, the strictness of the pathwidth hierarchy, the fact that the quasi-order is not a lattice, and the role of weakly sparse classes in the quasi-order.