On the first-order transduction quasiorder of hereditary classes of graphs

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures, and several important class properties can be defined in terms of transductions. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. This normal form allows to prove, among other results, that the local variants of (monadic) stability and (monadic) dependence are equivalent to their non-local versions. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of additive classes is also a bounded distributive join-semilattice. We characterize transductions of paths, cubic graphs, and cubic trees in terms of bandwidth, bounded degree, and treewidth. We establish that the classes of all graphs with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder and leave open whether the same holds for the classes of all graphs with treewidth at most $k$. We identify the obstructions for a class to be a transduction of a class with bounded degree, leading to an interesting transduction duality formulation. Eventually, we discuss a notion of dense analogs of sparse transduction-preserved class properties, and propose several related conjectures.