Fagin defined the class \(NP\) by the means of Existential Second-Order logic. Feder and Vardi expressed it (up to polynomial equivalence) by special fragments of Existential Second-Order logic (SNP), while the authors used forbidden expanded substructures (cf. lifts and shadows). Consequently, for such problems there is no dichotomy, unlike for CSPs.

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.

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex graphs.

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. 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. 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. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most k, for k ≥ 1 form a strict hierarchy in the FO transduction quasiorder.

We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. Thus, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context. We explore this emerging theory by proving the following: 1) A class of graphs is a first-order transduction of a class with bounded treewidth if and only if has bounded rankwidth and a stable edge relation (i.e. graphs from exclude some half-graph as a semi-induced subgraph). 2) If a class of graphs is monadically dependent and not monadically stable, then has in fact an unstable edge relation. As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly χ-bounded. Our proofs are effective and lead to polynomial time algorithms.

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths. These results show a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes: 1) Graphs with linear rankwidth at most r are linearly χ-bounded. Actually, they have bounded c-chromatic number, meaning that they can be colored with f (r) colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary family of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in . 3) For a class with bounded linear rankwidth the following conditions are equivalent: a) is stable, b) excludes some half-graph as a semi-induced subgraph, c) is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion , defined as first-order transductions of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth covers (or colorings), replacing treedepth by its dense analogue called shrubdepth.

The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful "depth" notion also for dense graphs (say; one which is stable under graph complementation)? To this end, in a 2012 conference paper, a new notion of shrub-depth has been introduced, such that it is related to the established notion of clique-width in a similar way as tree-depth is related to tree-width. Since then shrub-depth has been successfully used in several research papers. Here we provide an in-depth review of the definition and basic properties of shrub-depth, and we focus on its logical aspects which turned out to be most useful. In particular, we use shrub-depth to give a characterization of the lower ${\omega}$ levels of the MSO1 transduction hierarchy of simple graphs.

It is a well known experience that for sparse structures one can find fast algorithm for some problems which seem to be otherwise complex. The recently developed theory of sparse classes of graphs (and structures) formalizes this. Particularly the dichotomy Nowhere vs Somewhere Dense presents a very robust tool to study and design algorithms and algorithmic metatheorems. This dichotomy can be characterized in many different ways leading tp broad applications. We survey some of the recent highlights. This is a joint work with Patrice Ossona de Mendez (EHESS Paris and Charles University Prague).

Abstract A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

We prove that dual graphs and relational structures are connected. Moreover we give efficient bounds for their diameter: a linear bound in the case of oriented graphs (and this is best up to a constant) and a polynomial bound in the case of relational structures.

Classes of graphs with bounded expansion have been introduced in [15], [12]. They generalize both proper minor closed classes and classes with bounded degree.For any class with bounded expansion C and any integer p there exists a constant N(C,p) so that the vertex set of any graph G ∈ C may be partitioned into at most N(C,p) parts, any i ≤ p parts of them induce a subgraph of tree-width at most (i-1) [12] (actually, of tree-depth [16] at most i, what is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [14].We give here a simple algorithm to compute such partitions and prove that if we restrict the input graph to some fixed class C with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed C and p).This result is applied to get a linear time algorithm for the subgraph isomorphism problem with fixed pattern and input graphs in a fixed class with bounded expansion.More generally, let φ be a first order logic sentence. We prove that any fixed graph property of type "∃X: (|X| ≤ p) ⇿(G[X]=φ)" may be decided in linear time for input graphs in a fixed class with bounded expansion.

For a fixed countable homogeneous relational structure Γ we study the computational problem whether a given finite structure of the same signature homomorphically maps to Γ. This problem is known as the constraint satisfaction problem CSP(Γ) for the template Γ and has been intensively studied for finite Γ. We show that — as in the case of finite Γ — the computational complexity of CSP(Γ) for countable homogeneous Γ is determined by the clone of polymorphisms of Γ. To this end we prove the following theorem, which is of independent interest: the primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are preserved by the polymorphisms of Γ. If the age of Γ is given by a finite number of finite forbidden induced substructures, then CSP(Γ) is in NP. We use a classification result by Cherlin and prove that in this case every constraint satisfaction problem for a countable homogeneous digraph is either tractable or NP-complete.

Let ${\mathcal R}(n,k)$ denote the random k‐ary relation defined on the set $[n]=\{1,2,\dots,n\}$. We show that the probability that $([n], {\mathcal R}(n,k))$ is projective tends to one, as either n or k tends to infinity. This result implies that for most relational systems $(B,{{\underline{R}}})$ the ${{\textrm{CSP}}}(B,{{\underline{R}}})$ problem is NP‐complete (and thus that the dichotomy conjecture holds with probability 1), and confirms a conjecture of Rosenberg [I. G. Rosenberg, Rocky Mountain J. Math., 3 (1973), pp. 631–639].

We prove that there exists a constant k such that for every $n \geq 1$ there exists a directed core graph $H_n$ with at least $2^n$ vertices such that a directed graph G is $H_n$-colorable if and only if every subgraph of G with at most $kn\log(n)$ vertices is $H_n$-colorable. Our examples show that in general the "duals of relational structures" in the sense of [J. Nesetril and C. Tardif, J. Combin. Theory Ser. B, 80 (2000), pp. 80-97] can have superpolynomial size. The construction given in this paper gives a double exponential upper bound for such a construction. Here we improve this to an exponential upper bound.

Several classes of cyclic orders arising from geometrical, algebraical, and combinatorial structures are introduced, and their extendability to total cyclic orders is studied. By analogy to Dushnik–Miller dimension for partial orders we define, for circular orders, intersection and product dimension that may differ up to a factor of two. A class of cyclic orders that allow a graphic representation similar to Hasse diagrams is also studied.

The computational complexity of the following type of problems is studied. Given a topological layout (i.e., a drawing in the plane) of a graph, does it contain a noncrossing subgraph of a given type? It is conjectured that such problems are always NP-hard (provided planar subgraphs are looked for) regardless of the complexity of their nonplanar versions. This conjecture is verified for several cases in a very strong sense. In particular, it is shown that deciding the existence of a noncrossing path connecting two given vertices in a given topological layout of a 3-regular subgraph, as well as deciding the existence of a noncrossing cycle in such a layout, are NP-complete problems. It is also proved that deciding the existence of a noncrossing k-factor in a topological layout of a $( k + 1 )$-regular graph is NP-complete for $k = 2,3,4,5$. For $k = 1$, this question is NP-complete in layouts of 3-regular graphs, while it is polynomial solvable for layouts of graphs with maximum degree two.

We consider the set union problem (SUP) which consists in designing data for manipulation of a family of disjoint sets which partition a given universe of n elements. In response to a work of Tarjan we prove that the POSTORDER strategy for SUP has a linear length (thus solving a problem of Hart and Sharir). On the other side, we provide a data structure and axioms for an on line strategy-LOCAL POSTORDER-for SUP which fails to be linear but it has a very slow-indeed in the theory of finite sets unprovable-growth. This complements a result of Tarjan who showed an Ackermann type growth for a related problem. Our results may be summarized by saying that (in finite set theory) we may assume that our algorithms are linear (although we know that in fact they fail to be linear). Perhaps this is the first occurrence of unprovability in the complexity analysis of algorithms.