Lacon-, Shrub- and Parity-Decompositions: Characterizing Transductions of Bounded Expansion Classes

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts. In this paper, we show that one can encode graphs from a class with structurally bounded expansion via lacon-, shrub- and parity-decompositions from a class with bounded expansion. These decompositions are useful for lifting properties from sparse to structurally sparse graph classes.


Introduction
Many hard graph problems become easier if we assume the input to be well-structured. Usually, this is enforced by assuming that the input belongs to a certain kind of (infinite) graph class. This can, for example, be the class of all planar graphs or the class of all graphs with maximal degree at most d for some number d. Many of the established tractable graph classes are sparse in the sense that their graphs have relatively few edges compared to their number of possible edges.
Algorithmic meta-theorems state that whole families of problems become more tractable if we restrict the input to certain graph classes. The best known example is Courcelle's theorem [Cou90], which states that every problem definable in monadic second-order logic can be solved in linear time on bounded treewidth classes. It has been shown in a series of papers that all problems definable in first-order logic can be solved in linear time on bounded degree [See96], excluded minor [FG01a], locally bounded treewidth [FG01b] and more general sparse graph classes [DKT10,GKS17]. These results are obtained by solving the first-order model-checking problem on those graph classes. The input to this problem is a first-order sentence φ and a graph G and the task is to decide whether φ is satisfied on  complexity theoretic assumptions [CHKX06]. A first-order model-checking algorithm is considered efficient on some graph class if it solves the problem in time f (|φ|)|G| O(1) for some function f , i.e., in fpt time.
Nešetřil and Ossona de Mendez introduced bounded expansion and nowhere dense graph classes. These are very robust notions of sparsity that generalize the previously mentioned graph classes (see left side of Figure 1) and have many interesting algorithmic properties. Most notably, Dvořák, Král' and Thomas [DKT10] solve the first-order modelchecking problem in linear fpt time on bounded expansion classes, and Grohe, Kreutzer and Siebertz [GKS17] solve this problem in almost linear fpt time on nowhere dense graph classes. For sparse graphs, this is in a sense the best possible result of this type: If a graph class is monotone (i.e., closed under taking subgraphs), then the model-checking problem is fpt if and only if the class is nowhere dense (under standard complexity assumptions) . We therefore have reached a natural barrier in the study of meta-theorems for sparse graphs. But there are other well-structured graph classes that do not fit into the framework of sparsity. One of the current main goals in this area is to push the theory to account for dense (or non-monotone) classes as well. In particular, we want to find dense, but structurally simple graph classes on which one can solve the first-order model-checking problem in fpt time.
An established tool to capture such graph classes are first-order interpretations and transductions. For a given first-order formula φ(x, y), the corresponding (one dimensional 1 ) first-order interpretation is a function that maps an input a graph G to the graph I φ (G) with vertex set V (G) and edge set {uv | u, v ∈ V (G), u ̸ = v, G |= φ(u, v) ∨ φ(v, u)}. For example, with φ(x, y) = ¬edge(x, y), the interpretation complements the input graph. With φ(x, y) = ∃z edge(x, z) ∧ edge(z, y) it computes the square of a graph. For a given formula φ(x, y), the first-order interpretation of a graph class G is the graph class {I φ (G) | G ∈ G}. First-order transductions are slightly more powerful than interpretations, since they also have the ability to copy, delete and nondeterministically color vertices. As the precise definition of transductions is not important for now, we delay it until Section 2. We say a graph class G has structurally property X if it is the transduction of a graph class with property X (see right side of Figure 1 for examples). For the properties considered in this paper, this is equivalent to saying that G consists of induced subgraphs of an interpretation of a class with property X. This definition has nice closure properties in the sense that applying a transduction to a structurally property X class yields again a structurally property X class.
There have been recent efforts to lift the meta-theorems for sparse graph classes to transductions of these classes. Gajarský [DMS23], generalizing the previous results.
Such algorithmic results build upon structural decompositions of the corresponding graph classes. In this paper, we characterize structurally bounded expansion via lacondecompositions, shrub-decompositions and parity-decompositions. The techniques we introduce were later used to provide quasi-bush decompositions [DGK + 22] of similar flavor for structurally nowhere dense classes, which were in turn an important ingredient for the aforementioned model-checking algorithm on these classes [DMS23].
Before, bounded expansion classes have been characterized via low shrubdepth covers, where shrubdepth [dMON + 19, GHN + 12] can be understood as a dense equivalent to treedepth. This nicely complements the characterization of bounded expansion via low treedepth covers [NDM15].
We should mention another important approach to obtain dense, structurally simple graph classes. Bonnet, Kim, Thomassé and Watrigant [BKTW20] introduce a graph parameter called twinwidth that captures many interesting non-monotone graph classes. The authors provide an algorithm that -given an appropriate twinwidth contraction sequence -solves the first-order model-checking problem in linear fpt time [BKTW20]. Bounded twinwidth is orthogonal to structurally bounded expansion or structurally nowhere denseness. The notion of monadically NIP generalizes all aforementioned notions and is conjectured to precisely capture the hereditary tractable graph classes.
1.1. Introducing Lacon-, Shrub-and Parity-Decompositions. A lacon-, shrub-or parity-decomposition of a graph G is a labeled graph G ′ that encodes G in a possibly much sparser way. These decompositions are not tied to any fixed sparsity measure, instead one can characterize different dense graph classes by applying different sparsity requirements to G ′ . In this paper, we use them to capture structurally bounded expansion, structurally bounded treewidth and structurally bounded treedepth, i.e., shrubdepth (Theorem 1.6, 1.7, 1.8). Due to their flexibility, it seems quite likely that they can be used to capture transductions of other sparse classes as well. Generally speaking, we prove for certain properties X that a graph class has structurally property X if and only if each graph in the graph class has a lacon-, shrub-or parity-decomposition with property X.
A decomposition of a graph G consists of the so-called target vertices V (G) and additional hidden vertices. "Lacon" is an acronym for largest common neighbor, since there is an order on the hidden vertices and the label of the largest common neighbor of two vertices (either "0" or "1") determines whether they are adjacent or not. Formally, this is defined as follows. A linear order π on the vertices of a graph G is represented by an injective function π : V (G) → N. Definition 1.1 (Lacon-decomposition). A lacon-decomposition is a tuple (L, π) satisfying the following properties.
(1) L is a bipartite graph with sides T, H. We say T are the target vertices and H are the hidden vertices of the decomposition.
(3) π is a linear order on the vertices of L with π(h) < π(t) for all t ∈ T , h ∈ H.
is called the dominant vertex between t and t ′ . We say (L, π) is a lacon-decomposition of a graph G if (5) V (G) = T , (6) for all t ̸ = t ′ ∈ T there is an edge between t and t ′ in G if and only if the dominant vertex of t and t ′ is labeled with "1".
This definition leads to an illustrative process that reconstructs a graph G from its lacondecomposition (L, π): In the beginning, all edges in G are unspecified. Then we reveal the hidden vertices of L one by one in ascending order by π. If we encounter a hidden vertex h with label "1" we fully connect its neighborhood N (h) in G. On the other hand, if h has label "0" we remove all edges between vertices from N (h) in G. The edges inserted or removed by h "overwrite" the edges that were inserted or removed by a previous hidden vertex. After all hidden vertices have been revealed, the resulting graph is exactly G. See the middle Figure 2 for an example of a lacon-decomposition. Before we use lacon-decompositions to characterize transductions of different sparse graph classes, let us also introduce shrub-decompositions. Our definition closely follows the wording used by Ganian et al. [GHN + 12, dMON + 19] to define shrubdepth.
Definition 1.2 (Shrub-decomposition). Let m and d be nonnegative integers. A shrubdecomposition of m colors and diameter d of a graph G is a pair (F, S) consisting of an undirected graph F and a set S ⊆ {1, 2, . . . , m} 2 × {1, 2, . . . , d} (called signature) such that (1) the distance in F between any two vertices is at most d, (2) the set of pendant vertices (i.e., degree-one vertices) of F is exactly the set V (G) of vertices of G, (3) each pendant vertex of F is assigned one of the colors {1, 2 . . . , m}, and (4) for any i, j, l it holds (i, j, l) ∈ S iff (j, i, l) ∈ S (symmetry in the colors), and for any two vertices u, v ∈ V (G) such that u is colored with i and v is colored with j and the distance between u, v in F is l, the edge uv exists in G if and only if (i, j, l) ∈ S.
Item (4) says that the existence of a G-edge between u, v ∈ V (G) depends only on the colors of u, v and their distance in the decomposition. See the right side of Figure 2 for an example. A shrub-decomposition is a generalization of a tree-model used in the definition of shrubdepth. In fact, we can define shrubdepth (i.e., structurally bounded treedepth) using shrub-decompositions. Instead of requiring F to be a tree, we can require F to come from a bounded expansion class to obtain a characterization of structurally bounded expansion, as we will see soon. At last, we also introduce parity-decompositions, where adjacencies between vertices is encoded by the parity of the intersection of their neighborhoods in the decomposition.
Definition 1.4 (Parity-Decomposition). Let d be a nonnegative integer. A parity-decomposition of target-degree d is a bipartite graph P with sides T, H, where every vertex in T has degree at most d. We say T are the target vertices and H are the hidden vertices of the decomposition. We say P is a parity-decomposition of a graph G if V (G) = T , and for all t ̸ = t ′ ∈ T there is an edge between t and t ′ in G if and only if |N (t) ∩ N (t ′ )| is odd.

Generalized Coloring Numbers and Bounded Expansion.
Before we present our results, we introduce the so-called generalized coloring numbers, as we use them to pose a sparsity requirement on lacon-decompositions. These numbers can be used to characterize bounded expansion, treewidth and treedepth and have numerous algorithmic applications. Let G be an undirected graph and π be an ordering on the vertices of G. We say a vertex u is r-reachable from v with respect to π if π(u) ≤ π(v) and there is a path of length at most r from v to u and for all vertices w on the path either w = u or π(w) ≥ π(v). A vertex u is weakly r-reachable from v with respect to π if there is a path of length at most r from v to u and π(u) ≤ π(w) for all vertices w on that path. If G is a directed graph, we say u is (weakly) r-reachable from v if and only if it is (weakly) r-reachable in the underlying undirected graph. Let Reach r (G, π, v) be the set of vertices that are r-reachable from v with respect to π. We similarly define WReach r (G, π, v) be the set of weakly r-reachable vertices. We set col r (G, π) = max v∈V (G) |Reach r (G, π, v)|, wcol r (G, π) = max v∈V (G) |WReach r (G, π, v)|. We define Π (G) to be the set of all linear orders on G. Finally, the r-coloring number and weak r-coloring number of a graph G is defined as col r (G) = min π∈Π (G) col r (G, π), wcol r (G) = min π∈Π (G) wcol r (G, π).
For small r, the two flavors of generalized coloring numbers are strongly related. It holds that col r (G) ≤ wcol r (G) ≤ col r (G) r [KY03]. For large r, the generalized coloring numbers converge to treewidth (tw (G)) and treedepth (td(G)), respectively [NdM12, GKR + 18] col 1 (G) ≤ · · · ≤ col ∞ (G) = tw(G) + 1, A graph H is an r-shallow minor of a graph G if it is the result of first contracting mutually disjoint connected subgraphs with radius at most r in G and then taking a subgraph. A graph class G has bounded expansion if there exists a function f (r) such that |E(H)| |V (H)| ≤ f (r) for all G ∈ G and r-shallow minors H of G [NdM12]. Zhu first observed that generalized coloring numbers can also be used to characterize bounded expansion [Zhu09]. In this paper, we rely heavily on the following characterization by van den Heuvel and Kierstead [vK21], which is slightly stronger than Zhu's [Zhu09] original characterization.
Definition 1.5 (Bounded Expansion [vK21]). A graph class G has bounded expansion if there exists a function f (r) such that every graph G ∈ G has an ordering π of its vertices with col r (G, π) ≤ f (r) for all r.
As illustrated in Figure 1, examples of bounded expansion graph classes include planar graphs or any graph class of bounded degree. 1. 3. Main Result. This paper presents characterizations of structurally bounded expansion based on lacon-, shrub-and parity-decompositions. As a side result, we also obtain characterizations of structurally bounded treewidth and treedepth. For the definition of first-order transductions and structurally bounded expansion see Section 2. The following theorem contains the main contribution of this paper.
Theorem 1.6. Let G be a graph class. The following statements are equivalent.
(1) G has structurally bounded expansion, i.e., there exists a graph class G ′ with bounded expansion and a first-order transduction τ such that G ⊆ τ (G ′ ).
(2) There exists a function f (r) such that every graph in G has a lacon-decomposition (L, π) with col r (L, π) ≤ f (r) for all r (this implies that L comes from a bounded expansion class).
(3) There exist a signature S, a number d and a graph class G ′ with bounded expansion such that every graph in G has a shrub-decomposition (F, S) with one color, diameter at most d and F ∈ G ′ . (4) There exist number d and a graph class G ′ with bounded expansion such that every graph in G has a parity-decomposition P ∈ G ′ with target-degree at most d. Equivalently, structurally bounded expansion classes can also be characterized via low shrubdepth covers [GKN + 20]. This characterization was obtained using quantifier elimination, which is possible on bounded expansion [DKT13], but not on nowhere dense classes [GJdM + 21]. The techniques of this paper (local separators and logical composition theorems) have a better chance to generalize to structurally nowhere dense classes and beyond. In fact, in a follow-up paper [DGK + 22], we build upon this approach to derive shrub-decompositions as well as low shrubdepth covers for structurally nowhere dense graph classes.
Theorem 1.6 can further be understood as a limit on the expressive power of transductions on bounded expansion classes. For example, it implies that transductions based on boolean combinations of local, purely existential formulas have the same expressive power as general first-order transductions: Assume a graph class G was obtained from a graph class G ′ with bounded expansion via a (possibly very complicated) transduction. We replace G ′ with the class of shrub-decompositions of G, as described in (3). Now G can be expressed as a very simple transduction of G ′ . We merely have to check the color of two vertices and their (bounded) distance. This can be done using a boolean combination of local, existential formulas. When introducing function symbols f i (v) for the ith hidden-neighbor of a target vertex v in a lacon-decomposition, this further characterizes structurally bounded expansion classes as (functional) quantifier-free transductions of bounded expansion classes.
Other bounds on the expressive power of transductions on sparse graphs (orthogonal to ours) have been obtained in the literature [GKN + 20, NdMS20]. As side results, we obtain the following characterizations of structurally bounded treedepth and treewidth. We should note that structurally bounded treedepth is the same as bounded shrubdepth [GHN + 12].
Theorem 1.7. Let G be a graph class. The following statements are equivalent.
(1) G has structurally bounded treedepth, i.e., there exists a graph class G ′ with bounded treedepth and a first-order transduction τ such that G ⊆ τ (G ′ ).
(2) There exists a number d such that every graph in G has a lacon-decomposition (L, π) with wcol ∞ (L, π) ≤ d (this implies L has treedepth at most d).
(3) There exist a number d and a signature S such that every graph in G has a shrubdecomposition (F, S) with one color, diameter at most d and treedepth at most d. (4) There exist number d and a graph class G ′ with bounded treedepth such that every graph in G has a parity-decomposition P ∈ G ′ with target-degree at most d.
Theorem 1.8. Let G be a graph class. The following statements are equivalent.
(1) G has structurally bounded treewidth, i.e., there exists a graph class G ′ with bounded treewidth and a first-order transduction τ such that G ⊆ τ (G ′ ).
(2) There exists a number t such that every graph in G has a lacon-decomposition (L, π) with col ∞ (L, π) ≤ t − 1 (this implies L has treewidth at most t).
(3) There exist a number t and a signature S such that every graph in G has a shrubdecomposition (F, S) with one color, diameter at most t and treewidth at most t. (4) There exist number d and a graph class G ′ with bounded treewidth such that every graph in G has a parity-decomposition P ∈ G ′ with target-degree at most d.
The results of Theorem 1.6, 1.7, and 1.8 are consequences of a more general statement, Theorem 5.1, which shows that for every transduction of a graph G, we can find an equivalent lacon-decomposition whose generalized coloring numbers are not too far off from the numbers of G.

Localized Feferman-Vaught Composition Theorem.
The Feferman-Vaught theorem [Kar67] states that the validity of first-order formulas on the disjoint union or Cartesian product of two graphs is uniquely determined by the value of first-order formulas on the individual graphs. Makowsky adjusted the theorem for algorithmic use in the context of MSO model-checking [Mak04]. It has numerous applications and in the area of meta-theorems it leads, for example, to an especially concise proof of Courcelle's theorem [Gro08].
A highly useful property of first-order logic is locality, for example, in the form of Hanf's [Han65] or Gaifman's [Gai82] theorem. Intuitively, these theorems state that firstorder logic can only express local properties. Locality is a key ingredient in many first-order model-checking algorithms [DKR20, GKS17, KS17, See96, BDG + 22, DMS23].
A central building block in our proofs is a localized variant of the Feferman-Vaught theorem (also proved in [PST18, Lemma 15]). The essence of this result is the following: Assume we have a graph G, a first-order formula φ(x, y) and two vertices v, w and want to know whether G |= φ(v, w). Further assume that we have some kind of "local separator" between v, w, i.e., a tuple of verticesū such that all short paths between v and w pass throughū. We show that knowing whether G |= ψ(ū, v), G |= ψ(ū, w) for certain formulas ψ this gives us enough information to compute whether G |= φ(v, w). The original Feferman-Vaught theorem claims this only ifū is an actual separator between v and w, i.e., all paths between v and w pass throughū. To formalize our result, we need to define so-called q-types. The quantifier rank of a formula is the maximal nesting depth of its quantifiers.
We syntactically normalize formulas so that there are only finitely many formulas of fixed quantifier rank and with a fixed set of free variables. Therefore q-types are finite sets. For a tupleū = (u 1 , . . . , u k ), we denote the set {u 1 , . . . , u k } by set(ū). We follow Grohe's presentation of the Feferman-Vaught theorem [Gro08].
In the previous statement,ū separatesv andw by splitting G ∪ H into the subgraphs G and H. We extend this result using a more general notion of separation that goes as follows. The length of a path equals its number of edges.
Definition 1.11. Let G be a graph, r ∈ N, andū,v 1 , . . . ,v k be tuples of vertices from G. We sayū r-separatesv i andv j if every path of length at most r between a vertex fromv i and a vertex fromv j contains at least one vertex fromū. We sayū r-separatesv 1 , . . . ,v k if it r-separatesv i andv j for all i ̸ = j. In particular, if a vertex v appears in multiple tuples amongv 1 , . . . ,v k , then an r-separating tupleū also needs to contain v.
Based on this notion of separation, we use the following Feferman-Vaught-inspired result.
(i) We start with a graph G that is a transduction of a graph G ′ , where G ′ has bounded generalized coloring numbers. (i) =⇒ (ii) Then we construct a so-called directed lacon-decomposition (L, π) of G whose coloring numbers are bounded as well. This is a generalization of a lacondecomposition where we allow L to be a directed graph. This step forms the central part of the paper and here we use the localized Feferman-Vaught theorem extensively. (ii) =⇒ (iii) Next, we convert (L, π) into a normal lacon-decomposition without increasing the generalized coloring numbers too much. (iii) =⇒ (iv) We transform the lacon-decomposition into an equivalent shrub-decomposition (F, S), also with bounded coloring numbers. (iii) =⇒ (v) We transform the lacon-decomposition into an equivalent parity-decomposition P , also with bounded coloring numbers. (iv,v) =⇒ (i) This implies that G is a transduction of F or P , and since the generalized coloring numbers of both are bounded, this brings us back to the start. Afterwards, we use the fact that generalized coloring numbers can describe bounded expansion, bounded treewidth and bounded treedepth. Therefore, Theorem 1.6, 1.7 and 1.8 follow by posing different bounds on the coloring numbers.
The proof of (i) =⇒ (ii) is the central technical contribution of this paper. We sketch some its ideas. We have a graph G ′ with an ordering π of its vertices and want to obtain a lacon-decomposition of an interpretation I φ (G ′ ) of G ′ . Consider vertices v, w ∈ V (G ′ ) and a tupleū consisting of the vertices in WReach r (G ′ , π, v) ∩ WReach r (G ′ , π, w). It is a basic property of generalized coloring numbers that every path of length at most r between v and w passes throughū, i.e.,ū r-separates v and w. The localized decomposition theorem states that φ(v, w) depends only on q-types ofūv andūw. For every possible tupleū and every combination of q-types, we introduce a hidden vertex that we label with "1" if and only if its two q-types together imply φ to be true. We then connect the hidden vertices with the vertices of G ′ if their q-types match.
The implication (i) =⇒ (ii) is proved in Section 3 and (ii) =⇒ (iii) is proved in Section 4. This is then combined in Section 5 to show (i) =⇒ (iii). In Section 6 we prove (iii) =⇒ (iv) and in Section 7 we prove (iii) =⇒ (v). Then, in Section 8 we combine all these implications to prove our main results Theorem 1.6, 1.7 and 1.8. The localized Feferman-Vaught theorem has been proved before [PST18, Lemma 15]. We nevertheless finish the paper in Section 9 with a self-contained proof of the theorem.

Preliminaries
Any unusual graph notation is limited to specific sections and therefore introduced where it is used. We use first-order logic over labeled and unlabeled graphs. An unlabeled graph is a relational structure with a binary edge relation. A labeled graph may additionally have labels (or colors) on vertices or edges, represented by additional unary and binary relations. The length of a formula φ is denoted by |φ|. For a formula φ(x, y) and graph G, Transductions. First-order transductions extend interpretations with the ability to delete vertices, as well as copy and color the input graph. We use the same notation as [GHO + 16].
The central building block are basic transductions. A basic transduction is a triple τ 0 = (χ, ν, φ) of first-order formulas of arity zero, one and two.
There are two more building blocks, which are less important for this work. A p-parameter expansion is an operation that maps each graph G to the set of all graphs that can be created by adding p unary predicates (i.e., colors or labels) to

Interpretations of Bounded Expansion Have Directed Lacon-Decompositions
This section contains the central idea of this paper. We consider a generalization of lacondecompositions called directed lacon-decompositions and construct such a decomposition with bounded generalized coloring numbers. In the following definition we denote the inand out-neighborhoods of a vertex h by N − (h) and N + (h), respectively.
(1) L is a directed bipartite graph with sides T, H. We say T are the target vertices and H are the hidden vertices of the decomposition.
(3) π is a linear order on the vertices of L with π(h) < π(t) for all t ∈ T , h ∈ H.
is called the dominant vertex between t and t ′ . We say (L, π) is the directed lacon-decomposition of an undirected graph G if (5) V (G) = T , (6) for all t ̸ = t ′ ∈ T there is an edge between t and t ′ in G if and only if the dominant vertex of t and t ′ is labeled with "1".
We will need the following facts about generalized coloring numbers. As they can be easily derived from their basic definition, we omit a proof. Proposition 3.2. Let G be a graph with ordering σ and r ∈ N.
(2) Let v, w ∈ V (G) and S = WReach r (G, σ, v) ∩ WReach r (G, σ, w). Then every path of length at most r between v and w contains a vertex from S. Note that it is possible that v ∈ S or w ∈ S, in which case the statement holds trivially.
Lemma 3.3. Let φ(x, y) be a first-order formula. There exists a function g such that for every labeled graph G and ordering σ on the vertices of G there exists a directed lacon- Proof. For technical reasons, it is easier to prove the result of this lemma under the additional assumption that the input graph G has an apex vertex. We do so first, and at the end of the proof we generalize the result also to graphs without an apex vertex. Our main proof is outlined as follows: We first construct a directed lacon-decomposition, then prove that the construction encodes the graph correctly and at last bound the coloring numbers.
Constructing a Lacon-Decomposition. Let us fix a graph G (with an apex vertex) and an ordering σ. Let q = |φ|, l = wcol 2·4 q (G, σ) + 2 and let f (q, l) be the function from Theorem 1.12. We define the hidden vertices H of L to be all tuples (u,ū, type 1 , type 2 ) such that • u ∈ V (G), •ū is the tuple of all vertices in WReach 2·4 q (G, σ, u) ordered in ascending order by σ, • type 1 , type 2 are f (q, l)-types containing formulas with |ū| + 1 free variables.
Correctness of Construction. At first, we need to show that (L, π) is a directed lacondecomposition as defined in Definition 3.1. One can easily verify that L is in fact a directed bipartite graph, every hidden vertex is either labeled with "1" or "0" and that π is an ordering on the vertices of L with π(h) < π(t) for all h ∈ H, t ∈ T = V (G). What is left to do, is fix some vertices v 1 , v 2 ∈ T and show that the set is non-empty. We define R 1 = WReach 4 q (G, σ, v 1 ) and R 2 = WReach 4 q (G, σ, v 2 ). Since G has an apex vertex, there is a path of length at most two between v 1 and v 2 . We can also generally assume that 4 q ≥ 2 and therefore, by (2) of Proposition 3.2, R 1 ∩ R 2 ̸ = ∅. We choose some vertex u ′ ∈ R 1 ∩ R 2 and consider the hidden vertex h ′ = (u ′ ,ū ′ , tp f (q,l) (G,ū ′ v 1 ), tp f (q,l) (G,ū ′ v 2 )) whereū ′ is the tuple of all vertices in WReach 2·4 q (G, σ, u ′ ), ordered in ascending order by σ. We constructed L such that there is an arc from v 1 to h ′ and an arc from h ′ to v 2 . We conclude that is non-empty and therefore that (L, π) is a directed lacon-decomposition.
Next, we show that (L, π) also is a directed lacon-decomposition of I φ (G). L was constructed such that its target vertices are T = V (G) = V (I φ (G)). It remains to show that the dominant vertex of two arbitrary vertices v 1 , v 2 ∈ T is labeled with "1" if and only if v 1 v 2 ∈ E(I φ (G)). Let h be the dominant vertex of v 1 and v 2 . By construction, h is either of the form h = (u,ū, tp f (q,l) (G,ūv 1 ), tp f (q,l) (G,ūv 2 )) or h = (u,ū, tp f (q,l) (G,ūv 2 ), tp f (q,l) (G,ūv 1 )). W.l.o.g. we assume it is the former form.
Bounding the Coloring Numbers. For a hidden vertex (u,ū, type 1 , type 2 ) in L, we say u is its corresponding vertex in G. If t is a target vertex, we say the corresponding vertex in G is t itself. The corresponding vertex of a vertex x ∈ V (L) is denoted by u(x).
We start by bounding the number of hidden vertices that have the same corresponding vertex in G. To do so, we need to determine the number of possible types type 1 , type 2 a hidden vertex can have. The size ofū is bounded by wcol 2·4 q (G, σ). Thus, the number of f (q, l)-types containing formulas with at most |ū| + 1 free variables is bounded by some function of q and l. This means, there exists a function g ′ (independent of G and σ) such that Let h be a hidden vertex and h ′ ∈ Reach r (L, π, h) for some r. Since all target vertices are larger than h with respect to π, we know that h ′ is a hidden vertex. Remember that L is bipartite. By definition of the generalized coloring numbers, there is a path Here, the h i are hidden vertices and the t i are target vertices. The order π was constructed such that σ(u(h i )) > σ(u(h j )) implies π(h i ) > π(h j ) for all hidden vertices h i , h j . Therefore, σ(u(h 0 )) ≤ σ(u(h k )), and σ(u(h k )) = min k i=2 σ(u(h i )). We constructed L such that u(h i ), u(h i+1 ) ∈ WReach 4 q (G, σ, t i ). Thus by (1) of Proposition 3.2, either u(h i ) ∈ WReach 2·4 q (G, σ, u(h i+1 )) or u(h i+1 ) ∈ WReach 2·4 q (G, σ, u(h i )). By (3) of Proposition 3.2, there exists w ∈ Reach 4 q r (G, σ, u(h)) with u(h ′ ) ∈ WReach 2·4 q (G, σ, w). In other words, |Reach r (L, π, h)| is bounded from above by the number of tuples (w, v, h ′ ) with w ∈ Reach 4 q r (G, σ, u(h)), v ∈ WReach 2·4 q (G, σ, w) and u(h ′ ) = v. Using (3.1), we bound the number of such tuples by |Reach r (L, π, h)| ≤ col 4 q r (G, σ) · l · g ′ (q, l). (
The statement of this lemma also requires a bound on the weak coloring numbers wcol r (L, π). This bound can be proved in almost the same way as for col r (L, π) and therefore we only describe the main difference. Let h be a hidden vertex and h ′ ∈ WReach r (L, π, h).
No Apex Vertex. It remains to prove this result for the case that the input graph has no apex vertex. We reduce this case to the previously covered case with an apex. Let G be a graph without an apex and σ be an ordering of V (G). We construct a graph G ′ from G by adding an additional apex vertex and let σ ′ be the ordering of V (G ′ ) that preserves the order of σ but whose minimal element is the new apex vertex. Then col r (G ′ , σ ′ ) ≤ col r (G, σ) + 1 and wcol r (G ′ , σ ′ ) ≤ wcol r (G, σ) + 1. Assume we have a directed lacon-decomposition (L ′ , π ′ ) of I φ (G ′ ). We obtain a directed lacon-decomposition (L, π) of I φ (G) by removing the apex vertex from (L ′ , π ′ ). Then col r (L, π) ≤ col r (L ′ , π ′ ) and wcol r (L, π) ≤ wcol r (L ′ , π ′ ). These observations reduce the no-apex case to the apex case.

Directed and Undirected Lacon-Decompositions Have Same Expressive Power
In this section, we prove the following lemma, stating that directed and normal lacondecompositions are equally powerful.
Proof. Assume we have a directed lacon-decomposition (L, π) of a graph G with target vertices T and hidden vertices H. We construct an undirected lacon-decomposition (L ′ , π ′ ) of G with target and hidden vertices T, H ′ as follows: We start with H ′ = ∅. We process the vertices in H one by one in ascending order by π and add in each processing step various new vertices to H ′ . The ordering π ′ is thereby defined implicitly as follows (in ascending order): First come the vertices of H ′ in the order of insertion and then come the vertices of T in the same order as in π.
We will make sure that after a vertex h ∈ H has been processed the following invariant holds for all pairs of vertices v 1 ̸ = v 2 ∈ T . If v 1 , v 2 have a dominant vertex d in L such that π(d) ≤ π(h), then we guarantee that v 1 , v 2 have a dominant vertex d ′ in L ′ and that d ′ is labeled with "1" if and only if d is. After every vertex in H has been processed, this guarantees that (L ′ , π ′ ) is an undirected lacon-decomposition of G. While processing a vertex h ∈ H, certain new vertices are inserted into H ′ . The newly inserted vertices are in a sense either a copy of h or copies of vertices that have already been inserted into H ′ . These copies may later again be copied, and so forth. This leads to exponential growth. To keep track of these copies and to show that while being exponential, this grow is nevertheless bounded, every newly inserted vertex h ′ will have a memory, denoted by m(h ′ ). This memory consists of a sequence of vertices from H. If h ′ is derived from h, then we set m(h ′ ) = h. However, if h ′ is derived both from h and another vertex h ′′ that has previously been inserted into H ′ , then we set m(h ′ ) = m(h ′′ ) + h, i.e., m(h ′ ) is obtained by appending h to the memory of h ′′ . Let us now get into the details of the construction and prove our invariant.
Construction of Lacon-Decomposition. We describe the processing steps of a vertex h ∈ H. If N − (h)∪N + (h) = ∅ we do nothing. Otherwise, we add a vertex h ′ to H ′ that has the same label ("0" or "1") as h. We further add edges to We observe a problem.
Currently, h ′ is the dominant vertex for v 1 and v 2 in L ′ , even though h is not dominant for these vertices in L. Thus h ′ connects too many target vertices and we have to "undo" the effect of h ′ on the edges within the set N − (h) \ N + (h). The same holds for the set N + (h) \ N − (h). We do so as follows. We iterate over all vertices l ∈ Reach 2 (L ′ , π ′ , h ′ ) in order of insertion into H ′ . If These two vertices get the same label as l and since they were derived from both l and h, we set m(l ′ ) = m(l ′′ ) = m(l) + h. These new vertices l ′ and l ′′ undo the undesired effects of inserting h ′ , as we will prove soon. This completes the processing round of h. By induction, we know that the invariant was satisfied in the last round for all v 1 , v 2 . We show that it also holds after h is processed. Let v 1 ̸ = v 2 be two vertices with dominant vertex d in L such that π(d) ≤ π(h). We distinguish five cases.
In this case, neither h in (L, π) nor any of the newly inserted vertices in (L ′ , π ′ ) is dominant for v 1 , v 2 . Since the invariant for v 1 , v 2 held in the previous round, it also holds in this round. • v 1 ∈ N − (h) and v 2 ∈ N + (h). In this case, h and h ′ are dominant for v 1 , v 2 in (L, π) and (L ′ , π ′ ), respectively. The vertex h ′ is labeled with "1" if and only if h is. Thus, the invariant is fulfilled.
Then π(d) < π(h). Thus, in the previous round, there was a vertex that was dominant for v 1 , v 2 in (L ′ , π ′ ) that has the same label as d.
Since v 1 , v 2 ∈ N (h ′ ), this dominant vertex is contained in Reach 2 (L ′ , π ′ , h ′ ). In the construction, we iterate over all vertices l ∈ Reach 2 (L ′ , π ′ , h ′ ) in order of insertion into H ′ (i.e., ascending order by π ′ ) and insert vertices l ′ that satisfy v 1 , v 2 ∈ N (l ′ ) iff v 1 , v 2 ∈ N (l). The last vertex l with v 1 , v 2 ∈ N (l) that we encounter during this procedure was the dominant vertex of the previous round. The corresponding vertex l ′ has the same label as l and is the dominant vertex now.
Bounding the Coloring Numbers. For a vertex h ′ ∈ H ′ with m(h ′ ) = h 1 . . . h k , we say that h k is the corresponding vertex of h ′ in L. If t is a target vertex, we say the corresponding vertex in L is t itself. The corresponding vertex of a vertex x ∈ V (L ′ ) is denoted by u(x).
We fix a vertex h ∈ H and ask: How many h ′ ∈ H ′ can there be with u(h ′ ) = h? We pick a vertex h ′ ∈ H ′ with u(h ′ ) = h. Then it has a memory m(h ′ ) = h 1 . . . h k with h k = h. The following observations follow from the construction of (L ′ , π ′ ).
These three observations together imply that h 1 , . . . , h k ∈ Reach 2 (L, π, h). Thus m(h ′ ) consists of a subset of Reach 2 (L, π, h), written in ascending order by π. We can therefore bound the number of possible memories m(h ′ ) of h ′ by 2 col 2 (L,π) .
Furthermore, for each vertex l ∈ H ′ with memory l 1 . . . l k−1 and every vertex l k ∈ H there are at most two vertices (denoted by l ′ and l ′′ in the above construction) with the memory l 1 . . . l k . This means for a fixed memory of length k, there are at most 2 k vertices that have this particular memory. Combining the previous observations, we can conclude that for a fixed vertex h ∈ H there are at most 2 col 2 (L,π) · 2 col 2 (L,π) = 4 col 2 (L,π) vertices for all h ′ ∈ H ′ (see second item in enumeration above) implies the following for all x, y ∈ V (L ′ ): If x, y are adjacent in L ′ , then u(x), u(y) are adjacent in L. Also, the order π ′ was constructed such that π(u(x)) > π(u(y)) implies π ′ (x) > π ′ (y) for all x, y ∈ V (L ′ ). The last two points together mean • if y ∈ Reach r (L ′ , π ′ , x), then u(y) ∈ Reach r (L, π, u(x)) for all r, • if y ∈ WReach r (L ′ , π ′ , x), then u(y) ∈ WReach r (L, π, u(x)) for all r. Combining this with the observation that for a fixed u ∈ V (L) there are at most 4 col 2 (L,π) vertices y ∈ V (L ′ ) with u(y) = u gives us the bounds on col r (L ′ , π ′ ) and wcol r (L ′ , π ′ ) required by this lemma.

Transductions of Bounded Expansion Have Lacon-Decompositions
Now we combine the previous results into a more concise statement. We show that for every transduction of a graph G, we can find a lacon-decomposition whose coloring numbers are not too far off from the coloring numbers of G.
Theorem 5.1. Let τ be a transduction. There exist a constant c and a function f such that for every graph G, ordering σ on the vertices of G, and D ∈ τ (G) there exists a lacon-decomposition (L, π) of D with • col r (L, π) ≤ f (col c (G, σ)) · col cr (G, σ) for all r, • wcol r (L, π) ≤ f (col c (G, σ)) · wcol cr (G, σ) for all r.
All the key ideas needed to prove this result have already been established in Lemma 3.3 and Lemma 4.1. In this section, we merely combine them. First, we need to following technical but unexciting lemma.
Lemma 5.2. Let τ be a transduction. There exists a constant c and a basic transduction τ 0 such that for every graph G, ordering π on the vertices of G, and D ∈ τ (G) there exists a labeled graph G ′ with τ 0 (G ′ ) = D and ordering π ′ on the vertices of G ′ such that • col r (G ′ , π ′ ) ≤ c · col r (G, π) for all r, • wcol r (G ′ , π ′ ) ≤ c · wcol r (G, π) for all r.
Note that G ′ is the result of applying the copy operation ε and a coloring from γ to G. We extend π into an ordering π ′ on G ′ by placing the copied vertices next to the corresponding original vertices. If ε copies the graph c times, then col r (G ′ , π ′ ) ≤ c · col r (G, π) and wcol r (G ′ , π ′ ) ≤ c · wcol r (G, π). Now we prove Theorem 5.1 by chaining Lemma 5.2, 3.3 and 4.1 in this order.
Combining the previous 6 bounds then proves the result.

Converting Lacon-to Shrub-Decompositions
We now convert a lacon-decomposition into an equivalent shrub-decomposition while maintaining a bound on the generalized coloring numbers.
Proof. Let (L, π) be a lacon-decomposition of G. For every target vertex t and 1 ≤ i < |N (t)| we define ν i (t) to be the ith hidden neighbor in L, counted in descending order by π.
We construct a graph F from L (see Figure 3) by subdividing for every t ∈ V (G) and 1 ≤ i ≤ |N (t)| the edge between t and ν i (t) into a path of length We furthermore iteratively remove pendant vertices in V (F ) \ V (G) from F until the pendant vertices are exactly V (G).
Consider a path between two distinct vertices t 1 , t 2 ∈ V (G) in F . If it passes through two hidden vertices, then it has length at least 4 · 4col 1 (L, π), while a path passing through only one hidden vertex has length at most 2 · (4col 1 (L, π) + 2col 1 (L, π) + 1) = 12col 1 (L, π) + 2, which is shorter. Thus, the shortest path between any two vertices t 1 , t 2 ∈ V (G) in F passes through exactly one hidden vertex, namely their largest common neighbor in L. Each half of this path has even length iff the largest common neighbor is labeled with "0". Thus, two vertices are adjacent in G iff their distance in F modulo four is two.
Since our construction does not rely on any colors, we give every target vertex the same dummy color 1. The diameter of F is at most 12col 1 (L, π) + 2. We define the signature of the shrub decomposition as The bound on the coloring numbers of F can be proved as follows. Let H be the hidden vertices of L. Every connected component X of F [V (F ) \ H] consists of at most col 1 (L, π) many paths of length at most 6col 1 (L, π), intersecting in some shared vertex t ∈ V (G). Thus, X has size at most 7col 1 (L, π) + 1.
We extend the ordering π of L into an ordering π ′ of F by making the vertices in V (F )\V (L) larger than all vertices in V (L). Now for every h ∈ H we have Reach r (F, π ′ , h) ⊆ Reach r (L, π, h). For all other vertices v ∈ V (F )\H holds Reach r (F, π ′ , v) ⊆ Reach r (L, π, t)∪ X, where X is the connected component of v in F [V (F ) \ H] (of size at most 7col 1 (L, π) + 1) and t ∈ V (G) is the unique target vertex contained in X. An equivalent bound can be obtained for the weak reachability.

Converting Lacon-to Parity-Decompositions
The following proof is substantially similar to the one of Lemma 4.1.
Proof. Assume we have a lacon-decomposition (L, π) of a graph G with target vertices T and hidden vertices H. We construct a parity-decomposition P of G with target and hidden vertices T, H ′ as follows: We start with H ′ = ∅. We process the vertices in H one by one in ascending order by π and add in each processing step various new vertices to H ′ . This defines an ordering π ′ of V (P ) as follows (in ascending order): First come the vertices of H ′ in the order of insertion and then come the vertices of T in the same order as in π. Afterwards, we will use π to bound the coloring numbers of P .
We will make sure that after a vertex h ∈ H has been processed the following invariant holds for all pairs of vertices v 1 ̸ = v 2 ∈ T . If v 1 , v 2 have a dominant vertex d in L such that π(d) ≤ π(h) then we guarantee that |N (v 1 ) ∩ N (v 2 )| is odd in P if and only if d labeled with "1". After every vertex in H has been processed, this guarantees that P is a parity-decomposition of G. While processing a vertex h ∈ H, certain new vertices are inserted into H ′ . The newly inserted vertices are in a sense either a copy of h or copies of vertices that have already been inserted into H ′ . These copies may later again be copied, and so forth. This leads to exponential growth. To keep track of these copies and to show that while being exponential, this grow is nevertheless bounded, every newly inserted vertex h ′ will have a memory, denoted by m(h ′ ). This memory consists of a sequence of vertices from H. If h ′ is derived from h, then we set m(h ′ ) = h. However, if h ′ is derived both from h and another vertex h ′′ that has previously been inserted into H ′ , then we set m(h ′ ) = m(h ′′ ) + h, i.e., m(h ′ ) is obtained by appending h to the memory of h ′′ . Let us now get into the details of the construction and prove our invariant. Construction of Parity-Decomposition. We describe the processing steps of a vertex h ∈ H. If N L (h) = ∅ we do nothing. Otherwise iterate over all l ∈ H ′ with N P (l)∩N L (h) ̸ = ∅ in order of insertion to H ′ , and add a vertex l ′ to P with N P (l ′ ) = N P (l) ∩ N L (h) and m(l ′ ) = m(l) + h. Also, if h is labeled with "1", we add an additional vertex h ′ to P with N P (h ′ ) = N L (h) and m(h ′ ) = h. By induction, we know that the invariant was satisfied in the last round for all v 1 , v 2 . We show that it also holds after h is processed. Let v 1 ̸ = v 2 be two vertices with dominant vertex d in L such that π(d) ≤ π(h). We distinguish two cases.
• v 1 ̸ ∈ N L (h) or v 2 ̸ ∈ N L (h). In this case, h is not dominant for v 1 , v 2 in (L, π). Also none of the newly inserted vertices are contained in |N P (v 1 ) ∩ N P (v 2 )|. Since the invariant for v 1 , v 2 held in the previous round, it also holds in this round.
If h is labeled with "0", then |N P (v 1 ) ∩ N P (v 2 )| is even. This is because for every element l ∈ N P (v 1 ) ∩ N P (v 2 ) from the previous round, a "copy" l ′ with v 1 , v 2 ∈ N P (l ′ ) has been inserted. On the other hand,, if h is labeled with "1", the additional vertex Bounding the Coloring Numbers. For a vertex h ′ ∈ H ′ with m(h ′ ) = h 1 . . . h k , we say that h k is the corresponding vertex of h ′ in L. If t is a target vertex, we say the corresponding vertex in L is t itself. The corresponding vertex of a vertex x ∈ V (P ) is denoted by u(x).
We fix a vertex h ∈ H and ask: How many h ′ ∈ H ′ can there be with u(h ′ ) = h? We pick a vertex h ′ ∈ H ′ with u(h ′ ) = h. Then it has a memory m(h ′ ) = h 1 . . . h k with h k = h. The following observations follow from the construction of P .
These three observations together imply that h 1 , . . . , h k ∈ Reach 2 (L, π, h). Thus m(h ′ ) consists of a non-empty subset of Reach 2 (L, π, h), written in ascending order by π. We can therefore bound the number of possible memories m(h ′ ) of h ′ by 2 col 2 (L,π) − 1. Also, every vertex h ′ ∈ H ′ is uniquely identified by its memory m(h ′ ) and every vertex t ∈ T has u(t) = t. We can therefore conclude for for every h ∈ H that there are at most 2 col 2 (L,π) vertices h ′ ∈ V (P ) with u(h ′ ) = h.
The fact that N P (h ′ ) ⊆ N L u(h ′ ) for all h ′ ∈ H ′ (see second item in enumeration above) implies the following for all x, y ∈ V (P ): If x, y are adjacent in P , then u(x), u(y) are adjacent in L. Also, the order π ′ was constructed such that π(u(x)) > π(u(y)) implies π ′ (x) > π ′ (y) for all x, y ∈ V (P ). The last two points together mean • if y ∈ Reach r (P, π ′ , x), then u(y) ∈ Reach r (L, π, u(x)) for all r, • if y ∈ WReach r (P, π ′ , x), then u(y) ∈ WReach r (L, π, u(x)) for all r.
Combining this with the observation that for a fixed u ∈ V (L) there are at most 2 col 2 (L,π) vertices y ∈ V (P ) with u(y) = u gives us the bounds on col r (P, π ′ ) and wcol r (P, π ′ ) required by this lemma. Since every target vertex in L has degree at most col 1 (L, π), P has targetdegree at most 2 col 2 (L,π) · col 1 (L, π).
Proof. Our result follows from proving the following claim via structural induction. Let φ(x,ȳ 1 , . . . ,ȳ k ) be a first-order formula with quantifier rank q. Then one can compute a boolean combination Φ(x,ȳ 1 , . . . ,ȳ k ) of first-order formulas of the form ξ(x,ȳ i ) such that for all graphs G and all tuplesv 1 , . . . ,v k that are 4 q -separated by a tupleū in G holds We call such a boolean combination Φ a separated expression. The claim holds for atomic formulas becauseū 4 0 -separatesv 1 , . . . ,v k , i.e., there are no edges between set(v i ) \ set(ū) and set(v j ) \ set(ū) in G for i ̸ = j. It also holds for boolean combinations and negations. It remains to prove our claim for the case that φ(x,ȳ 1 , . . . ,ȳ k ) = ∃xψ for some formula ψ.
For t ∈ N and v ∈ V (G), we define Nū t (v) to be the set of all vertices in G that are reachable from v via a path of length at most t that contains no vertex fromū. For a tuplev, we define Nū t (v) = v∈set(v) Nū t (v). We write z ∈ Nx t (ȳ) as a short-hand for a first-order formula ν t (x,ȳ, z) such that for all graphs G, vertex-tuplesū,v and vertices w holds G |= ν t (ū,v, w) iff w ∈ Nū t (v). The formula ν t (x,ȳ, z) can be constructed using t − 1 existential quantifiers.
We will proceed by finding separated expressions for φ * and for each φ i . Since 4 q = 4r, we consider a graph G and tuplesv 1 , . . . ,v k that are 4r-separated by a tupleū in G. We start with φ i . This formula asks whether there exists an x ∈ Nx r (ȳ i ) satisfying ψ. For every u ∈ Nū r (v i ) holds thatū r-separates the tuplesv i u andv j for i ̸ = j, since otherwise there would be a short path fromv i via u tov j that contains no vertex fromū. By the induction hypothesis, there exists a separated expression Ψ i (x,ȳ 1 , . . . ,ȳ i x, . . .ȳ k ) such that for all u ∈ Nū r (v i ) G |= ψ(ū, u,v 1 , . .  Next, we want to obtain a separated expression for φ * . Let therefore u ∈ V (G) with u ̸ ∈ Nū r (v i ) for all i. Then the tuples u,v 1 , . . . ,v k are r-separated byū. By the induction hypothesis, there exists a separated expression Ψ * (x, x,ȳ 1 , . . . ,ȳ k ) such that G |= ψ(ū, u,v 1 , . . . ,v k ) ⇐⇒ G |= Ψ * (ū, u,v 1 , . . . ,v k ).
We can assume Ψ * to be of the form Note that Φ * is not yet a separated expression. Nevertheless, substituting Φ i and Φ * into equation (9.1) yields Nū r (v i ) ∧ ξ l (ū, x) k j=1 ξ lj (ū,v j ) .
To complete the proof, it is sufficient to find a separated expression equivalent to γ. We therefore define the separated expression Γ(ū,v 1 , . . . ,v and make a case distinction based on it.
We further define for 0 ≤ h ≤ k i=1 |v i | a first-order formula δ h (x) which is true iff s R(x) > h. This formula can be constructed in a similar way as the formulas δ ih (x,ȳ i ) above. We use these formulas to construct the separated expression ∆(x,ȳ 1 , . . . ,ȳ k ) = with G |= ∆(ū,v 1 , . . . ,v k ) ⇐⇒ k i=1 s R(ū) ∩ Nū r (v i ) < s R(ū) . This completes our decomposition of φ into separated expressions.