Twin-width and permutations
?douard Bonnet, Jaroslav Nešet?il, Patrice Ossona de Mendez, Sebastian Siebertz, Stéphan Thomassé
TTwin-width and permutations
Édouard Bonnet ! ˇ Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Jaroslav Nešetřil ! Computer Science Institute of Charles University (IUUK), Praha, Czech Republic
Patrice Ossona de Mendez ! ˇ Centre d’Analyse et de Mathématique Sociales CNRS UMR 8557, Franceand Computer Science Institute of Charles University (IUUK), Praha, Czech Republic
Sebastian Siebertz ! University of Bremen, Bremen, Germany
Stéphan Thomassé ! Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Abstract
Inspired by a width invariant defined on permutations by Guillemot and Marx, the twin-widthinvariant has been recently introduced by Bonnet, Kim, Thomassé, and Watrigant. We provethat a class of binary relational structures (that is: edge-colored partially directed graphs) hasbounded twin-width if and only if it is a first-order transduction of a proper permutation class. Asa by-product, it shows that every class with bounded twin-width contains at most 2 O ( n ) pairwisenon-isomorphic n -vertex graphs. Theory of computation → Finite Model Theory; Mathematics ofcomputing → Graph theory
Keywords and phrases
Twin-width, first-order transductions, structural graph theory
Funding
This paper is part of a project that has received fundingfrom the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovationprogramme (grant agreement No 810115 –
Dynasnet ). Many constructions of graphs with good structural and algorithmic properties are based ontrees. In turn this often leads to important parameters and to a hierarchy of graph properties(and graph classes), which allow to treat hard problems in a parametric way. Examplesinclude tree-depth, treewidth, cliquewidth, shrub-depth to name just a few. These notionshave relevance and in many cases are a principal tool for important complexity and algorith-mic results: Let us mention at least Rossman’s homomorphism preservation theorem [18],Courcelle’s theorem on MSO definable properties [8, 9], Robertson and Seymour’s graphminor theory [17], and low tree-depth decompositions of classes with bounded expansion [14].In this paper we consider the twin-width graph parameter, defined by Bonnet, Kim,Thomassé and Watrigant [6] as a generalization of a width invariant for classes of permutationsdefined by Guillemot and Marx [11]. This parameter was intensively studied recently inthe context of many structural and algorithmic questions such as FPT model checking [6],graph enumeration [4], graph coloring [3], and matrices and ordered graphs [5]. We showthat twin-width can be concisely expressed by special structures called here twin models.Twin models are rooted trees augmented by a set of transversal edges that satisfies two a r X i v : . [ c s . L O ] F e b Twin-width and permutations simple properties: minimality and consistency. These properties imply that every twin modeladmits rankings, from which we can compute a width. The twin-width of a graph thencoincides with the optimal width of ranked twin-model of the graph. While this connectionis technical, twin models provide a simple way to handle classes with bounded twin-width.We show that a class C of binary relational structures has bounded twin-width if andonly if C is a transduction of a proper permutation class (i.e., a class of permutations closedunder sub-permutations, which avoids at least one pattern). The involved transductionbeing k -bounded, it is then a consequence of [13] that any class of relational structures withbounded twin-width contains at most c n non-isomorphic structures with n vertices hence issmall (i.e., contains at most c n n ! labeled structures with n vertices). This extends the mainresult of [4] while not using the “versatile contraction” machinery (but only the preservationof bounded twin-width by transduction proved in [6]). This also extends a similar propertyfor proper minor-closed classes of graphs, which can be derived from the boundedness ofbook thickness, as noticed by Colin McDiarmid (see the concluding remarks of [2]). Thissuggests that every small hereditary class of graphs contains at most c n non-isomorphicgraphs with n vertices (see Conjecture 8.1).This paper is a combination of model-theoretic tools (relational structures, interpretations,transductions), structural graph theory and theory of permutations. The fact that any classof graphs with bounded twin-width is just a transduction of a proper permutation class issurprising at first glance and it nicely complements another model theoretic characterizationof these classes: a class of graphs has bounded twin-width if and only if it is the reduct of adependent class of ordered graphs [5]. On the other hand twin-models are interesting objectsper se and in a way presents one of the weakest forms of width parameters related to trees.Note also that for other classes of sparse structures we have no such a concrete model.We outline the proof of our main result, by relying on Figure 1. The relevant terminologywill be formally introduced in the appropriate sections. O L (cid:43) (cid:43) (cid:43) (cid:43) ⇌ Y < O (cid:107) (cid:107) (cid:107) (cid:107) F (2-bounded) S (cid:126) (cid:126) (cid:126) (cid:126) T L (cid:43) (cid:43) (cid:43) (cid:43) ⇌ Reduct (cid:79) (cid:79) (cid:79) (cid:79) T < T O (cid:107) (cid:107) (cid:107) (cid:107) T G (cid:43) (cid:43) (cid:43) (cid:43) Reduct (cid:15) (cid:15) (cid:15) (cid:15)
Reduct (cid:79) (cid:79) (cid:79) (cid:79) ⇌ D < T U (cid:107) (cid:107) (cid:107) (cid:107) T U (cid:43) (cid:43) (cid:43) (cid:43) ⇌ Reduct (cid:15) (cid:15) (cid:15) (cid:15) E < T G (cid:107) (cid:107) (cid:107) (cid:107) Reduct (cid:15) (cid:15) (cid:15) (cid:15) T (cid:43) (cid:43) (cid:43) (cid:43) ⇌ P T (cid:107) (cid:107) (cid:107) (cid:107) C twin-model (cid:47) (cid:47) T full twin-model (cid:90) (cid:90) Gaifman (cid:43) (cid:43) (cid:43) (cid:43) ⇌ D Unfold (cid:107) (cid:107) (cid:107) (cid:107)
Unfold (cid:43) (cid:43) (cid:43) (cid:43) ⇌ E Gaifman (cid:107) (cid:107) (cid:107) (cid:107)
Figure 1
Relations between the classes of structures involved in the proof of the main result.
We start with a class C of binary relational structures with bounded twin-width. Wederive a class T of twin-models (tree-like representations of the graphs using rooted binarytrees and transversal binary relations). Replacing the rooted binary trees of the twin-modelsby binary tree-orders, we get a class F of so-called full twin-models, which we prove hasbounded twin-width. This class can be used to retrieve C as a transduction, that is by means . Bonnet, J. Nešetřil, P. Ossona de Mendez, S. Siebertz, and S. Thomassé 3 of a logical encoding. Using a transduction pairing (generalizing the notion of a bijectiveencoding) between binary tree-orders ( O ) and rooted binary trees ordered by a preorder ( Y < )we derive a transduction pairing of F with a class T < of ordered twin-models. From theproperty that the class D of the Gaifman graphs of the twin-models in T is degenerate (andhas bounded twin-width) we prove a transduction pairing of T and D , from which we derivea transduction pairing of T < and the class D < of ordered Gaifman graphs of the orderedtwin-models. From a transduction pairing of D with a class E of binary structures, in whicheach binary relation induces a pseudoforest, we deduce a transduction pairing of D < with anexpansion E < of E by a linear order. We then prove a transduction pairing of this class witha class P of permutations. As this class has bounded twin-width (as it is a transduction ofa class with bounded twin-width) we infer that P avoids a least one pattern. Following thebackward transductions, we eventually deduce that C is a transduction of the hereditaryclosure of P , which is a proper permutation class. We assume basic knowledge of first-order logic and refer to [12] for extensive background.A relational signature
Σ is a finite set of relation symbols R i with associated arity r i .A relational structure A with signature Σ, or simply a Σ -structure consists of a domain A together with relations R i ( A ) ⊆ A r i for each relation symbol R i ∈ Σ with arity r i . Therelation R i ( A ) is called the interpretation of R i in A . We will often speak of a relation insteadof a relation symbol when there is no ambiguity. We may write A as ( A, R ( A ) , . . . , R s ( A )).In this paper we will consider relational structures with finite domain, and (mostly) withrelations of arity at most 2. Without loss of generality we assume that all structures containat least two elements. We will further assume that Σ-structures are irreflexive , that is,( v, v ) ̸∈ R i ( A ) for every element v ∈ A and relation symbol R i ∈ Σ. A unary relation is calleda mark . Let R be a binary relation symbol and let u, v ∈ A . That the pair ( u, v ) lies in theinterpretation of R in A will be indifferently denoted by ( u, v ) ∈ R ( A ) or A | = R ( u, v ). Moregenerally, for a formula φ ( x , . . . , x k ), a Σ-structure A , an integer ℓ < k and a , . . . , a ℓ ∈ A we define φ ( A , a , . . . , a ℓ ) := { ( x , . . . , x k − ℓ ) ∈ A k − ℓ : A | = φ ( x , . . . , x k − ℓ , a , . . . , a ℓ ) } . Let A = ( A, R ( A ) , . . . , R s ( A )) be a Σ-structure and let X ⊆ A . The substructure of A induced by X is the Σ-structure A [ X ] = ( X, R ( A ) ∩ X r , . . . , R k ( A ) ∩ X r s ). Graphs are structures with a single binary relation E encoding adjacency; this relation isanti-reflexive and symmetric. Graphs of particular interest in this paper are rooted trees.For a rooted tree Y , we denote by I ( Y ) the set of internal nodes of Y , by L ( Y ), the set ofleaves of Y , by r ( Y ), the root of Y , and by ⪯ Y , the partial order on V ( Y ) defined by u ⪯ Y v if the unique path in Y linking r ( Y ) and v contains u (i.e., if u = v or u is an ancestor of v in Y ). For a non-root vertex v , we further denote by π Y ( v ) the parent of v , which is theunique neighbor of v smaller than v with respect to ⪯ Y . A rooted binary tree is a rootedtree such that every internal node has exactly two children. Partial orders are structures with a single anti-symmetric and transitive binary relation ≺ .Particular partial orders will be of interest here. Linear orders are partial orders such that ∀ x ∀ y ( x ≺ y ∨ y ≺ x ∨ y = x ). Tree orders are partial orders that satisfy the followingaxioms: ∀ x ∀ y ∀ z (cid:0) ( x ≺ z ∧ y ≺ z ) → ( x ≺ y ∨ y ≺ x ∨ x = y ) (cid:1) and ∃ r ∀ x ( x = r ∨ r ≺ x ). Itwill be convenient to use ⪯ , ≻ , ⪰ with their obvious meaning. Let ( X, ≺ ) be a tree-order. Twin-width and permutations
The infimum inf( u, v ) of two elements u, v ∈ X is the unique element w ∈ X such that w ⪯ u, w ⪯ v , and ∀ z (cid:0) ( z ⪯ u ∧ z ⪯ v ) → z ⪯ w (cid:1) . Note that inf( x, y ) is first-order definablefrom ≺ , hence can be used in our formulas. A binary tree order is a tree-order ( X, ≺ ) thatsatisfies the axiom ∀ x ∃ y ∃ y (cid:0) y ̸ = y ∧ ∀ z ( z ≻ x ) → ( z ⪰ y ∨ z ⪰ y ) (cid:1) . Ordered graphs are structures with two binary relations, E and < , where E defines agraph and < defines a linear order. We denote ordered graphs as G < = ( V, E, < ).A permutation is represented as a structure σ = ( V, < , < ), where V is a finite setand where < and < are two linear orders on this set (see e.g. [7, 1]). Two permutations σ = ( V, < , < ) and σ ′ = ( V ′ , < ′ , < ′ ) are isomorphic if there is a bijection between V and V ′ preserving both linear orders. Let X ⊆ V . The sub-permutation of σ inducedby X is the permutation on X defined by the two linear orders of σ restricted to X . Theisomorphism types of the sub-permutations of a permutation σ are the patterns of σ . Aclass P of (isomorphism types of) permutations is hereditary (or closed ) if it is closed undertaking sub-permutations. A permutation class is a hereditary class of permutations. Apermutation class is proper if it is not the class of all permutations. Note that the terms“class of permutations” and “permutation class” are not equivalent, the second referring to ahereditary class of permutations, as it is customary. Let Σ , Σ ′ be signatures. A (simple) interpretation I of Σ ′ -structures in Σ-structures is definedby a formula ρ ( x ), and a formula ρ R ′ ( x , . . . , x k ) for each k -ary relation symbol R ′ ∈ Σ ′ (where, by formula, we mean a first-order formula in the language of Σ-structures). Let I be an interpretation of Σ ′ -structures in Σ-structure, where Σ ′ = { R ′ , . . . , R ′ s } . For eachΣ-structure A we denote by I ( A ) = ( ρ ( A ) , ρ R ′ ( A ) , . . . , ρ R ′ s ( A )) the Σ ′ -structure interpretedby I in A . Similarly, for a class C we denote by I ( C ) the set { I ( A ) : A ∈ C } .We denote by Reduct Σ + → Σ (or simply Reduct when Σ and Σ + are clear from context) theinterpretation that “forgets” the relations in Σ + \ Σ while preserving all the other relationsand the domain. For a Σ + -structure B , the Σ-structure Reduct ( B ) is called the Σ- reduct (or simply reduct if Σ is clear from the context) of B . A class C is a reduct of a class D if C = Reduct ( D ). Conversely, a class D is an expansion of C if C is a reduct of D .Another important interpretation is Gaifman Σ (or simply Gaifman when Σ is clear fromcontext), which maps a Σ-structure A to its Gaifman graph , whose vertex set is A and whoseedge set is the set of all pairs of vertices included in a tuple of some relation.Note that an interpretation of Σ -structures in Σ -structure naturally defines an inter-pretation of Σ +2 -structures in Σ +1 -structures if Σ +2 \ Σ = Σ +1 \ Σ by leaving the relationsin Σ +1 \ Σ unchanged (that is, by considering ρ R ( x , . . . , x k ) = R ( x , . . . , x k ) for theserelations). Let Σ , Σ ′ be signatures. A simple transduction T from Σ-structures to Σ ′ -structures is definedby a simple interpretation I T of Σ ′ -structures in Σ + -structures, where Σ + is a signatureobtained from Σ by adding finitely many marks. For a Σ-structure A , we denote by T ( A ) theset of all I T ( B ) where B is a Σ + -structure with reduct A : T ( A ) = { I T ( B ) : Reduct ( B ) = A } .Let k ∈ N . A k -blowing of a Σ-structure A is the Σ ′ -structure B = A • k , where Σ ′ isthe signature obtained from Σ by adding a new binary relation ∼ encoding an equivalencerelation. The domain of A • k is B = A × [ k ], and, denoting p the projection A × [ k ] → A we have, for all x, y ∈ B , B | = x ∼ y if p ( x ) = p ( y ), and (for R ∈ Σ) B | = R ( x , . . . , x k ) if . Bonnet, J. Nešetřil, P. Ossona de Mendez, S. Siebertz, and S. Thomassé 5 A | = R ( p ( x ) , . . . , p ( x k )). A copying transduction is the composition of a k -blowing and asimple transduction; the integer k is the blowing factor of the copying transduction T and isdenoted by bf( T ). It is easily checked that the composition of two copying transductionsis again a copying transduction. In the following by the term transduction we mean acopying transduction. Note that for every transduction T from Σ-structure to Σ ′ , for everyΣ-structure A and for every Σ ′ -structure B ∈ T ( A ) we have | B | ≤ bf( T ) | A | .Let T , T ′ be transductions from Σ-structures to Σ ′ -structures, and let C be a class ofΣ-structures. The transduction T ′ subsumes the transduction T on C if T ′ ( A ) ⊇ T ( A ) for all A ∈ C . If C is a class of Σ-structures we define T ( C ) = S A ∈ C T ( A ). We say that a class D of Σ ′ -structure is a T -transduction of C if D ⊆ T ( C ) and, more generally, the class D is a transduction of the class C , and we write C (cid:47) (cid:47) (cid:47) (cid:47) D , if there exists a transduction T suchthat D is a T -transduction of C . Note that we only require the inclusion of D in T ( C ). Theclass D is a c -bounded T -transduction of the class C if, for every B ∈ D there exists A ∈ C with B ∈ T ( A ) and | A | ≤ c | B | . Two classes C and D are transduction equivalent if each isa transduction of the other. A transduction pairing of two classes C and D is a pair ( D , C )of (copying) transductions, such that ∀ A ∈ C ∃ B ∈ D ( A ) ∩ D A ∈ C ( B ) and ∀ B ∈ D ∃ A ∈ C ( B ) ∩ C B ∈ D ( A ) . We denote by C (cid:41) (cid:41) (cid:41) (cid:41) ⇌ D (cid:105) (cid:105) (cid:105) (cid:105) the existence of a transduction pairing of C and D . Remark thatif ( C , D ) is a transduction pairing then C is bf( D )-bounded and D is bf( C )-bounded. Inspired by a width invariant defined on permutations by Guillemot and Marx [11], thetwin-width invariant tww has been recently introduced by Bonnet, Kim, Thomassé, andWatrigant [6]. Graph twin-width is originally defined using a sequence of near-twin identifi-cations. The twin-width of the graph intuitively measures the accumulated errors (kept trackof by so-called red edges ) made by the identifications. Our definition for binary relationalstructures will be equivalent when restricted to undirected graphs, but will slightly differfrom the definition of the twin-width of binary structures (via matrix encodings) given in [6],though linearly tied. Note that forbidding unary relations does not harm in our context, astransductions can freely reintroduce any number of unary relations.In order to define twin-width, we first need to introduce some preliminary notions, whichgeneralize the notion of trigraphs (i.e., graphs with some red edges) introduced in [6]. Let Σbe a binary relational signature. The signature Σ ∗ is obtained by adding, for each binaryrelation symbol R a new binary relation symbol R ∗ . The symbol R ∗ will always be interpretedas a symmetric relation and plays for R the role of red edges in [6].Let A be a Σ ∗ -structure and let u and v be vertices of A . The vertices u, v are R -clones fora vertex w and a relation R ∈ Σ if we have A | = (cid:0) R ( u, w ) ↔ R ( v, w ) (cid:1) ∧ (cid:0) R ( w, u ) ↔ R ( w, v ) (cid:1) and no pair in R ∗ contains both w and either u or v . The Σ ∗ -structure A ′ obtained by contracting u and v into a new vertex z is defined as follows: A ′ = A \ { u, v } ∪ { z } ; R ( A ′ ) ∩ ( A ′ \ { z } ) × ( A ′ \ { z } ) = R ( A ) ∩ ( A \ { u, v } ) × ( A \ { u, v } ) for all R ∈ Σ ∗ ;for every vertex w ∈ A ′ \ { z } and every R ∈ Σ such that u and v are R -clones for w , welet A ′ | = R ( w, z ) if A | = R ( u, z ) and A ′ | = R ( z, w ) if A | = R ( z, u );otherwise, for every vertex w ∈ A ′ \ { z } and every R ∈ Σ such that u and v are not R -clones for w we let A ′ | = R ∗ ( w, z ) ∧ R ∗ ( z, w ). Twin-width and permutations A d -sequence for a Σ-structure A is a sequence A n , . . . , A of Σ ∗ -structures such that: A n is isomorphic to A ; A is the Σ ∗ -structure with a single element; for every 1 ≤ i < n , A i is obtained from A i +1 by performing a single contraction; for every 1 ≤ i < n and every v ∈ A i , the sum of the degrees in relations R ∗ ∈ Σ ∗ \ Σ of v in A i is less or equal to d (thedegree of v in relation R ∗ is defined as the degree of the undirected graph ( A, R ∗ ( A )). Theminimum d such that there exists a d -sequence for a Σ-structure A is the twin-width tww( A )of A . A crucial property of twin-width is the following result. ▶ Theorem 2.1 ([6]) . Let C , D be classes of binary structures. If C has bounded twin-widthand D is a transduction of C , then D has bounded twin-width. Sparse classes of graphs have many nice properties. The next property will be of particularinterest here. Recall that a star coloring of a graph G is a proper coloring of G such thatany two color classes induce a star forest (i.e., a disjoint union of stars); the star chromaticnumber χ st ( G ) of G is the minimum number of colors in a star coloring of G . Note thata star coloring of a graph with c defines a partition of the edge set into (cid:0) c (cid:1) star forests.Although we are interested only in binary relational structures in this paper, the next lemmaholds (and is proved) for general relational signatures. ▶ Lemma 3.1.
Let Σ be a relational signature, let C be a class of Σ -structures, and let c bean integer. There exists a simple transduction Unfold c from graphs to Σ -structures such that ifgraphs in C have star chromatic number at most c , then ( Unfold c , Gaifman ) is a transductionpairing of ( C , Gaifman ( C )) . Proof.
Let c = sup { χ st ( G ) : G ∈ Gaifman ( C ) } < ∞ . Let A ∈ C , let G = Gaifman ( A ), andlet γ : V ( G ) → [ c ] be a star coloring of G . In G , any two color classes induce a star forest,which we orient away from their centers. This way we get an orientation ⃗G of G such thatfor every vertex v and every in-neighbor u of v , the vertex u is the only in-neighbor of v with color γ ( u ). Let R ∈ Σ be a relation of arity k . For each ( u , . . . , u k ) ∈ R ( A ), u , . . . , u k induce a tournament in ⃗G . Every tournament has at least one directed Hamiltonian path [16].We fix one such Hamiltonian path and let p ( u , . . . , u k ) be the index of the last vertex in thepath. Let a = p ( u , . . . , u k ), let ( c , . . . , c k ) = ( γ ( u ) , . . . , γ ( u k )). Then there exists exactlyone clique of size k containing u a with vertices colored c , . . . , c k . Indeed, let ( u i , . . . , u i k )be the Hamiltonian path associated to the tournament induced by u , . . . , u k in ⃗G . Then u i k = u a and, for each 1 ≤ j < k the vertex u i j is the unique neighbor of u i j +1 with color c i j .For each relation R ∈ Σ with arity k and each ( u , . . . , u k ) ∈ R ( A ) we put at v = u p ( u ,...,u k ) a mark M Rγ ( u ) ,...,γ ( u k ) . We further put at each vertex v a mark C γ ( v ) . Then the structure A is reconstructed by the transduction Unfold c defined by the formulas ρ R ( x , . . . , x k ) := _ c ,...,c k (cid:16) ^ ≤ j ≤ k C c i ( x i ) ∧ ^ ≤ i In this section we formalize the notions of twin-models and ranked twin-models, which arereminiscent of the “ordered union trees” and “interval biclique partitions” adopted in [3].This structure will allow to encode a contraction sequence and to give an alternative definitionof twin-width. ▶ Definition 4.1 (twin-model) . Let Σ = ( R , . . . , R k ) be a binary relational signature.A Σ-twin-model (or simply a twin-model when Σ is clear from the context) is a tuple ( Y, Z R , . . . , Z R k ) where Y is a rooted binary tree and each Z R i is a binary relation satisfyingthe following minimality and consistency conditions: (minimality) if ( u, v ) ∈ Z R i , then there exists no ( u ′ , v ′ ) ̸ = ( u, v ) with u ′ ⪯ Y u , v ′ ⪯ Y v and ( u ′ , v ′ ) ∈ Z R i ; (consistency) if a traversal of a cycle γ in Y ∪ S i Z R i traverses all the Y -edges (of γ )away from the root, then γ contains two consecutive edges in S i Z R i .A twin-model ( Y, Z R , . . . , Z R k ) defines the Σ -structure A (or ( Y, Z R , . . . , Z R k ) is a twin-model of A ) if A = L ( Y ) and, for each R i ∈ Σ , R i ( A ) is the set of all pairs ( u, v ) suchthat there exists u ′ ⪯ Y u and v ′ ⪯ Y v with ( u ′ , v ′ ) ∈ Z R i . ▶ Definition 4.2 (ranking, boundaries, and layers) . Let ( Y, Z R , . . . , Z R k ) be a twin-model ofa Σ -structure A with | A | = n . A ranking τ of the twin-model ( Y, Z R , . . . , Z R k ) is a mappingfrom V ( Y ) to [ n ] that satisfies the following labeling, monotonicity, and synchronicityconditions: (labeling) the function τ restricted to I ( Y ) is a bijection with [ n − , and is equal to n on L ( Y ) ; (monotonicity) If u ≺ Y v , then τ ( u ) < τ ( v ) ; (synchronicity) If ( u, v ) ∈ Z R i , then max( τ ( π Y ( u )) , τ ( π Y ( v ))) < min( τ ( u ) , τ ( v )) .A ranked twin-model is a tuple T = ( Y, Z R , . . . , Z R k , τ ) , where ( Y, Z R , . . . , Z R k ) is atwin-model, and τ is a ranking of ( Y, Z R , . . . , Z R k ) .For < t ≤ n , the boundary ∂ t Y is the set ∂ t Y = { u ∈ V ( Y ) | τ ( u ) ≥ t ∧ τ ( π Y ( u )) < t } and the layer L t is the Σ ∗ -structure with vertex set ∂ t Y and relations defined by R i ( L t ) = { ( u, v ) ∈ ∂ t Y × ∂ t Y | ∃ u ′ ⪯ Y u, ∃ v ′ ⪯ Y v, ( u ′ , v ′ ) ∈ Z R i } R ∗ i ( L t ) = { ( u, v ) ∈ ∂ t Y × ∂ t Y | ∃ u ′ ⪰ Y u, ∃ v ′ ⪰ Y v, ( u ′ , v ′ ) ̸ = ( u, v ) and { ( u ′ , v ′ ) , ( v ′ , u ′ ) } ∩ Z R i ̸ = ∅} . For t = 1 we define the boundary ∂ Y = { r ( Y ) } and the layer L as the Σ ∗ -structure withunique vertex r ( Y ) . Twin-width and permutations ▶ Definition 4.3 (width) . The width of a ranked twin-model T = ( Y, Z R , . . . , Z R k , τ ) isdefined as width( T ) = max t ∈ [ n ] max v ∈ L t X R i ∈ Σ | R ∗ i ( L t , v ) | . At first sight the consistency condition of a twin-model (of A ) may seem contrived.One may for instance wonder if the minimality and consistency conditions are not simplyequivalent to the property that every ( u, v ) ∈ R i ( A ) is realized by a unique unordered pair u ′ , v ′ with ( u ′ , v ′ ) ∈ Z R i , u ′ ⪯ Y u , and v ′ ⪯ Y v . In case the structure A encodes a simpleundirected graph G (with signature Σ = ( E )), we would simply impose that the edges of Z E partition the edges of G into bicliques.In Figure 2 we give a small example that shows that this property is not strong enoughto always yield a ranking. This illustrates why the consistency condition is what we want(no more, no less) and also serves as a visual support for the notions of contraction sequence,twin-model, and ranking. ab cdef 215 43 a b c d e f a b c d e f ? ? ? Figure 2 Left: A 6-vertex graph and a contraction sequence, where the tiny digit in each boxindicates the number of remaining vertices when the vertex appears. Center: A twin-model ofthe graph, where the edges of Z E are in bold blue, and a ranking (for the internal nodes) of thistwin-model that actually matches the contraction sequence. Right: A flawed twin-model where theedge set E is indeed partitioned by the pairs of Z E . Here no ranking is possible: By the symmetry,one just needs to consider the labeling of the parent of a, b with 5, and that then the edges bc and bd cannot be realized. There is indeed a cycle b ? d ? f ? with all the tree arcs oriented the same way,and without two consecutive edges of Z E . On the contrary all such cycles in the central tree havetwo consecutive edges of Z E , like 5 bd f has ( b, d ) , ( d, ∈ Z E . In this section we prove that every d -contraction sequence of a Σ-structure A defines aranked twin-model of A with width at most d .A d -sequence A n , . . . , A for a Σ-structure A defines a rooted binary tree Y with vertexset V ( Y ) = S i A i and set of leaves L ( Y ) = A n as follows: for each i ∈ [ n − 1] let z i be thevertex of A i and u i , v i be the vertices of A i +1 such that z i results from the contraction of u i and v i in A i +1 . Then I ( Y ) = { z i : i ∈ [ n − } , r ( Y ) = z , and the children of z i in Y arethe vertices u i and v i . . Bonnet, J. Nešetřil, P. Ossona de Mendez, S. Siebertz, and S. Thomassé 9 For each relation R ∈ Σ we define a binary relation Z R on V ( Y ) as follows. Let z i be thevertex of A i resulting from the contraction of u i and v i in A i +1 . If ( u i , v i ) ∈ R ( A i +1 ), then( u i , v i ) ∈ Z R . If u i and v i are not R -clones for w , then the pairs involving w and u i or v i in R ( A i +1 ) are copied in Z R . Intuitively, Z R collects the R -relations when they just appear(in the order A , . . . , A n ). We further define Z = S R ∈ Σ Z R and the function τ : V ( Y ) → [ n ]by τ ( v ) = n if v ∈ L ( Y ) and τ ( z i ) = i . Note that for each i ∈ [ n ] and non-root vertex v of Y ,we have v ∈ A i if and only if τ ( π Y ( v )) < i ≤ τ ( v ). ▶ Lemma 4.4. Every d -sequence A n , . . . , A defines a ranked twin-model with width atmost d . Proof. ▷ Claim 4.5. The function τ satisfies the labeling, monotonicity, and synchronicity conditions. Proof. The first two conditions are straightforward. Let ( u, v ) ∈ Z R . Let i ∈ [ n − 1] be suchthat ( u, v ) appears in A i for the first time. As u, v ∈ A i we have both τ ( π Y ( u )) < i ≤ τ ( u )and τ ( π Y ( v )) < i ≤ τ ( v ), i.e., the synchronicity condition holds. ◁▷ Claim 4.6. The relations Z R ( R ∈ Σ) satisfy the minimality and consistency conditions. Proof. The minimality condition follows directly from the definition. Let ⃗H be the orientedgraph obtained from Y by orienting all the edges from the root and adding, for each R ∈ Σand each pair ( u, v ) ∈ Z R the arcs π Y ( u ) v and π Y ( v ) u whenever they do not exist. It followsfrom the monotonicity and synchronicity conditions that ⃗H is acyclically oriented. Indeed,any arc ( x, y ) in ⃗H satisfies τ ( x ) < τ ( y ).Assume towards a contradiction that in Y ∪ S R ∈ Σ Z R one can find a cycle γ such thatthe orientation of the Y -edges is consistent with a traversal of γ and γ does not contain twoconsecutive edges in S R ∈ Σ Z R . By replacing in γ each group formed by an edge in S R ∈ Σ Z R and its preceding edge in Y by the corresponding arc in ⃗H we obtain a circuit in ⃗H ,contradicting its acyclicity. Hence the relations Z R satisfy the consistency condition. ◁ The next claim is immediate from the definition and ends the proof of the lemma. ▷ Claim 4.7. The ranked twin-model ( Y, Z R , . . . , Z R k , τ ) derived from a d -sequence A n , . . . , A has width at most d . ◀ In this section we establish two properties of twin-models. The first one is the equality ofthe minimum width of a twin-model with the twin-width of a structure; the second one isthat twin-models of structures with bounded twin-width are degenerate. ▶ Lemma 4.8. Every twin-model has a ranking, and the twin-width of a Σ -structure A isthe minimum width of a ranked twin-model of A . Proof. The following claim, which asserts that no Z R i “crosses” the boundaries, will be quitehelpful. ▷ Claim 4.9. Let t ∈ [ n − 1] and let u, v ∈ ∂ t Y . Then there exists no pair ( u ′ , v ′ ) ∈ Z R i with u ′ ≺ Y u and v ′ ≻ Y v . Proof. Assume ( u ′ , v ′ ) ∈ Z R i and v ′ ≻ Y v . By the synchronicity property we have τ ( u ′ ) >τ ( π Y ( v ′ )) ≥ τ ( v ) ≥ t , contradicting τ ( u ′ ) ≤ τ ( π Y ( u )) < t . ◁ For a Σ ∗ -structure A and R ∈ Σ we define R ( A ) = { ( u, v ) ∈ A : { ( u, v ) , ( v, u ) } ∩ ( R ( A ) ∪ R ∗ ( A )) ̸ = ∅} . ▷ Claim 4.10. Let L , . . . , L n be the layers of a ranked tree model T of a Σ-structure A .Then there exists a contraction sequence A n , . . . , A of A with A i = L i and, for each R ∈ Σ, R ( A i ) = R ( L i ), R ( L i ) ⊆ R ( A i ), and R ∗ ( L i ) ⊇ R ∗ ( A i ). Proof. For i ∈ [ n − A i is obtained from A i +1 by contracting the pairof vertices u i , v i into w i , where w i is the vertex of Y with τ ( w i ) = i and u i and v i arethe two children of w i in Y . It is easily checked that A i = L i . Let z be a vertex of A i different from w i . Then ( z, w i ) ∈ R ( A i ) if there exists a leaf w ′ ⪰ Y w i and a leaf z ′ ⪰ Y z such that { ( w ′ , z ′ ) , ( z ′ , w ′ ) } ∩ R ( A ) ̸ = ∅ . As T is a twin-model of A this means that thereexists w ′′ ⪯ Y w ′ and z ′′ ⪯ Y z ′ with ( w ′′ , z ′′ ) ∈ Z R or ( z ′′ , w ′′ ) ∈ Z R . As ⪯ Y is a tree-order, w i and w ′′ are comparable, as well as z and z ′′ . From this and Claim 4.9 it follows { ( z, w i ) , ( w i , z ) } ⊆ R ( L i ).We now prove R ( L i ) ⊆ R ( A i ) by reverse induction on i . For i = n we have R ( L i ) = R ( A i ) = R ( A ). Let i ∈ [ n − 1] and let u i , v i , w i be defined as above. If ( w i , z ) ∈ R ( L i ), thenthere exists w ′ ⪯ Y w i and z ′ ⪯ Y z with ( w ′ , z ′ ) ∈ Z R thus we have also ( u i , z ) ∈ R ( L i +1 )and ( v i , z ) ∈ R ( L i +1 ). By induction we deduce ( u i , z ) ∈ R ( A i +1 ) and ( v i , z ) ∈ R ( A i +1 ).Similarly, if ( z, w i ) ∈ R ( L i ), then ( z, u i ) ∈ R ( A i +1 ) and ( z, v i ) ∈ R ( A i +1 ). Thus u i and v i are R -clones for z hence if ( w i , z ) ∈ R ( L i ), then ( w i , z ) ∈ R ( A i ) and if ( z, w i ) ∈ R ( L i ), then( z, w i ) ∈ R ( A i ). It follows that we have R ( L i ) ⊆ R ( A i ). Thus we have R ∗ ( L i ) = R ( L i ) \ { ( u, v ) : { ( u, v ) , ( v, u ) } ∩ R ( L i ) = ∅}⊇ { ( u, v ) : R ( A i ) \ { ( u, v ) : { ( u, v ) , ( v, u ) } ∩ R ( A i ) = ∅} = R ∗ ( A i ) . ◀ We are now able to prove the first part of the statement. ▷ Claim 4.11. Every twin-model has a ranking. Proof. Consider the oriented graph ⃗H obtained from orienting Y from the root and adding,for each R ∈ Σ and each pair ( u, v ) ∈ Z R , an arc π ( u ) v and an arc π ( v ) u (whenever they donot exist). Assume for contradiction that ⃗H contains a directed circuit. Replace each arc ofthe form π ( u ) v of this circuit (with ( u, v ) ∈ Z R ) by the path ( π ( u ) u, uv ) in the twin-model.This way we obtain a closed walk in Y ∪ S R ∈ Σ Z R traversing all edges of Y away from theroot and no two consecutive edges are in S R ∈ Σ Z R , contradicting the consistency assumption.Thus ⃗H is acyclic and a topological ordering of ⃗H [ I ( Y )] extends to a labeling τ : V ( H ) → [ n ]that is bijective between I ( Y ) and [ n − n on L ( Y ), and increasing with respect toevery arc of ⃗H . This directly implies both the monotonicity and the synchronicity properties. ◁ We are now able to complete the proof of the lemma. According to Lemma 4.4, every d -sequence for A defines a ranked twin-model with width at most d . Conversely, every rankedtwin-model for A with width d ′ defines a sequence of layers L t with max v ∈ L t P R i ∈ Σ | R ∗ i ( L t , v ) |≤ d ′ and, by Claim 4.10, a d ′ -sequence for A . ◀ The following easy remark will be useful. . Bonnet, J. Nešetřil, P. Ossona de Mendez, S. Siebertz, and S. Thomassé 11 ▷ Claim 4.12. Let Y = ( Y, Z R , . . . , Z R k , τ ) be a ranked twin-model of a Σ-structure A andlet X ⊆ A . Let Y ′ be the subtree of Y induced by all the vertices in X and their ancestorsin Y , let Z ′ R i be the subset of all pairs in Z R i ∩ ( Y ′ × Y ′ ), and let τ ′ be the mapping from Y ′ to [ | X | ] such that for every x, y ∈ V ( Y ′ ) we have τ ( x ) < τ ( y ) ⇐⇒ τ ′ ( x ) < τ ′ ( y ). Then Y ′ = ( Y ′ , Z ′ R , . . . , Z ′ R k , τ ′ ) is a ranked twin-model of A [ X ], whose width is not larger thanthe one of Y . ▶ Lemma 4.13. The Gaifman graph of a twin-model of a Σ -structure with width d is d + k + 1 -degenerate, where k = | Σ | . Proof. Let A = ( A, R ( A ) , . . . , R k ( A )) be a Σ-structure, let T = ( Y, Z R , . . . , Z R k , τ ) be aranked twin-model of A with width d , and let G be the Gaifman graph of ( Y, Z R , . . . , Z R k ).The ranked twin-model T (with layers L i ) defines a d -sequence A n , . . . , A , where A i = L i (see Lemma 4.8). Let z be the node with τ ( z ) = n − u and v be its children. Eachpair in Z R i containing u (except pairs containing to both u and v ) gives rise (in A n − ) toan R ∗ i -edge incident to z when contracting u and v . Thus the degree of u in G is at most d + k + 1 ( d for the sum of the degrees in the relations R ∗ i , k for the pairs ( u, v ) ∈ Z R i , and 1for the tree edge ( u, z ). Then, in G − u , the vertex v has also degree at most d + k + 1. Nowwe remark that removing u and v from Y , and redefining τ ( x ) as min( n − , τ ( x )), we get aranked twin-model of A n − (minus R ∗ i -edges) with width at most d , whose Gaifman graphis G − u − v . By induction, we deduce that G is d + k + 1-degenerate. ◀ To reconstruct a Σ-structure A from a twin-model ( Y, Z R , . . . , Z R k ), we make use of the tree-order ⪯ Y defined by Y . As this tree-order cannot be obtained as a first-order transduction of( Y, Z R , . . . , Z R k ) it will be convenient to introduce a variant of twin-models: the full twin-model associated to a twin-model ( Y, Z R , . . . , Z R k ) is the structure ( V ( Y ) , ≺ Y , Z R , . . . , Z R k ).Let S be the simple interpretation of Σ-structures in full twin-models defined by formulas ρ ( x ) := ¬ ( ∃ y y ≺ Y x ); ρ R i ( x, y ) := ∃ u ∃ v ( u ⪯ Y x ) ∧ ( v ⪯ Y y ) ∧ Z R i ( u, v ) . The following claim follows directly from the definition of a twin-model. ▷ Claim 5.1. If T = ( X, ≺ , Z R , . . . , Z R k ) is a full twin-model of A , then S ( T ) = A and | A | = ( | T | + 1) / ▶ Lemma 5.2. Let T = ( Y, Z R , . . . , Z R k , τ ) be a ranked twin-model, with associated fulltwin-model T = ( V ( Y ) , ≺ Y , Z R , . . . , Z R k ) . Then the width of T is at most twice the widthof ( V ( Y ) , ≺ Y , Z R , . . . , Z R k , τ ) . Proof. Let I , I be copies of I ( Y ) and let p i : I ( Y ) → I i be the “identity” for i = 0 , 1. Wedefine the binary rooted tree b Y with vertex set V ( b Y ) = V ( Y ) ∪ I ∪ I , leaf set L ( b Y ) = V ( Y ),root r ( b Y ) = p o ( r ( Y )), and parent function π b Y ( x ) = p ◦ π Y ( x ) if x ∈ L ( Y ) p ( x ) if x ∈ I ( Y ) p ◦ p − ( x ) if x ∈ I p ◦ π Y ◦ p − ( x ) if x ∈ I \ { r ( b Y ) } x if x = r ( b Y ) An informal description of b Y is that it is obtained by replacing every internal node v of Y bya cherry C v (i.e., a complete binary tree on three vertices) such that one leaf of C v remainsa leaf in b Y , the other leaf of C v is linked to the “children of v ,” while the root of C v is linkedto the “parent of v ” (provided v is not the root of Y ).We further define b Z ≺ = { ( v, p ( v )) : v ∈ I ( Y ) } . ▷ Claim 5.3. ( b Y , b Z ≺ , Z R , . . . , Z R k ) is a twin-model of T . Proof. We have V ( Y ) = L ( b Y ). The minimality condition are obviously satisfied for b Z ≺ and Z R i . Let b Z = b Z ≺ ∪ S i Z R i . Consider a cycle b γ in b Y ∪ b Z , with all the edges in b Y orientedaway from the root. Assume for contradiction that no two edges in b Z are consecutive in b γ .Then either b γ contains a directed path of b Y linking to vertices in L ( Y ) or a directed pathof b Y linking a vertex in I ( Y ) to a distinct vertex in V ( Y ). As no such directed paths existin b Y we are led to a contradiction. Hence ( b Y , b Z ≺ , Z R , . . . , Z R k ) satisfies the consistencycondition.For u, v ∈ V ( Y ) there exists u ′ ⪯ b Y u and v ′ ⪯ b Y v with ( u ′ , v ′ ) ∈ Z R i if and only if( u, v ) ∈ Z R i . For u, v ∈ V ( Y ) there exists u ′ ⪯ b Y u and v ′ ⪯ b Y v with ( u ′ , v ′ ) ∈ b Z ≺ if and onlyif u ′ = u, v ′ = p ( u ), and v ′ ⪯ b Y v , that is, if and only if u ≺ Y v . Hence ( b Y , b Z ≺ , Z R , . . . , Z R k )is a twin-model of T . ◁ Let n = | L ( Y ) | . The next claim shows that we have much freedom in defining a rankingfunction for ( b Y , b Z ≺ , Z R , . . . , Z R k ). ▷ Claim 5.4. If ˆ τ : V ( b Y ) → [2 n − 1] satisfy the labeling and monotonicity conditions, thenˆ τ is a ranking of ( b Y , b Z ≺ , Z R , . . . , Z R k ). Proof. Assume ( u, v ) ∈ b Z ≺ . Then π b Y ( u ) = π b Y ( v ) hence the synchronicity for b Z ≺ followsfrom monotonicity. Assume ( u, v ) ∈ Z R i . Then ˆ τ ( u ) = ˆ τ ( v ) = 2 n − ◁ We now define ˆ τ : V ( b Y ) → [2 n − 1] as follows: order the vertices v ∈ I by increasing τ ◦ p − ( v ). For each v ∈ I , insert the children of v in I just after v , then add r ( b Y ) in thevery beginning. Numbering the vertices of I ∪ I according this order defines ˆ τ on I ( b Y ).We extend this function to the whole V ( b Y ) by defining ˆ τ ( v ) = 2 n − v ∈ L ( b Y ).By construction, the labeling and monotonicity properties hold hence ˆ τ is a ranking of( b Y , b Z ≺ , Z R , . . . , Z R k ).Consider a time 1 < ˆ t < n − v be the vertex with ˆ τ ( v ) = ˆ t . If v ∈ I we define t = τ ◦ p − ( v ). Then ∂ ˆ t b Y = p ( ∂ t Y ) and the degree for Z ∗ R i in the layer of b Y at time ˆ t is atmost the degree for Z ∗ R i in the layer of Y at time t and the degree for Z ∗≺ is null. If v ∈ I we define t = τ ◦ π Y ◦ p − ( v ). Then ∂ ˆ t b Y is p ( ∂ t Y ) in which we remove the parent of v andadd v and (maybe) the sibling of v . Compared to the layer at time ˆ τ ◦ π ( π Y ◦ p − ( v )), thered degree can only increase because some Z ∗ R i from a vertex u are adjacent to v and itssibling. It follows that the maximum Z ∗ R i is at most doubled. ◀ In this section, a transduction pairing of binary tree-orders and preordered rooted binarytrees will allow use to consider ordered twin-models (having a sparse reduct) instead of fulltwin-models. . Bonnet, J. Nešetřil, P. Ossona de Mendez, S. Siebertz, and S. Thomassé 13 ▶ Lemma 6.1. Let O be the class of binary tree orders and let Y < be the class of rootedbinary trees ordered by some preorder. Then there exists simple transductions L and O suchthat ( L , O ) is a transduction pairing of O and Y < . Proof. We define two simple transductions. The first transduction maps binary tree-orders ≺ into the preorder defined by some traversal of Y . L is defined as follows: we consider a mark M on the vertices and define ρ E ( x, y ) := (cid:0) ( x ≺ y ) ∧ ∀ v ¬ ( x ≺ v ∧ v ≺ y ) (cid:1) ∨ (cid:0) ( y ≺ x ) ∧ ∀ v ¬ ( y ≺ v ∧ v ≺ x ) (cid:1) ρ < ( x, y ) := ( x ≺ y ) ∨ ¬ ( y ⪯ x ) ∧ ∃ u ∃ v ∃ w (cid:16) ∀ z (cid:0) w ≺ z → ¬ ( z ≺ u ∧ z ≺ v ) (cid:1) ∧ ( u ⪯ x ∧ v ⪯ y ∧ w ≺ u ∧ w ≺ v ∧ ρ E ( u, w ) ∧ ρ E ( v, w ) ∧ M ( u ) (cid:17) . Consider a binary tree order ( V ( Y ) , ≺ ) ∈ O , and let Y be the rooted binary tree definedby ≺ . Recall that the preorder of Y defined by some plane embedding of Y (that is toan ordering, for each node v , of the children of v ) is a linear order on V ( Y ) such that forevery internal node v of Y , one finds in the ordering the vertex v , then the first childrenof v and its descendants, then the second children of v and its descendants. Let < be thepreorder defined by some plane embedding of Y . Let us mark by M all the nodes of Y that are the first children of their parent. The formula ρ E defines the cover graph of ≺ ,thus E is the adjacency relation of Y . Let x, y be nodes of Y . If x = y , then ρ < ( x, y ) doesnot hold. If x and y are comparable in ≺ , then ρ < ( x, y ) is equivalent to x ≺ y . Otherwise,let w be the infimum of x and y , and let u and v be the children of z such that u ≺ x and v ≺ y . Then ρ < ( x, y ) holds if u is the first children of w , that is if w is marked. Altogether,we have ρ < ( x, y ) if and only if x < y . Hence ( V ( Y ) , ≺ ) ∈ L ( Y < ), where Y < stands for Y ordered by < .The transduction O is defined as follows: ρ ≺ ( x, y ) := ( x < y ) ∧ ∀ z ∀ w ( x < z ∧ z ≤ y ∧ E ( z, w )) → ( x ≤ w ) . Let Y < be a rooted binary tree Y with preorder < and let ≺ be the corresponding tree-order.If x ≥ y , then ρ ≺ ( x, y ) does not hold so we assume x < y . Assume x is an ancestor of y in Y , then all the vertices z between x and y in the preorder are descendants of x thus anyneighbour of these are either descendants of x or x itself thus ρ ≺ ( x, y ) holds. Otherwise,let w be the infimum of x and y in Y and let z be the children of z that is an ancestor of y .Then z is between x and y is the preorder, z is adjacent to w , but w appears before x in thepreorder. Thus ρ ≺ ( x, y ) does not hold. It follows that ρ ≺ ( x, y ) is equivalent to x ≺ y thus Y < ∈ O ( V ( Y ) , ≺ ). Thus ( L , O ) is a transduction pairing of O and Y < . ◀▶ Lemma 6.2. Let C be a class of Σ -structures with bounded twin-width, and let T be aclass of optimal twin-models of the Σ -structures in C . Then there exist a simple transduction Unfold such that ( Gaifman , Unfold ) is a transduction pairing of T and Gaifman ( T ) . Proof. Let F be the class of full twin-models of the Σ-structures in C derived from thetwin-models in C . According to Lemma 5.2 the class F has bounded twin-width. Let T < bethe class of ordered twin-models obtained from the twin-models in T by adding a linear ordergiven by the preorder of the rooted tree of the tree model. Then T < is an L -transductionof F hence has bounded twin-width. Thus the class T , being a reduct of T < , has boundedtwin-width. It follows from Lemma 4.13 that the class Gaifman ( T ) is degenerate, hence,according to Theorem 3.2, it has bounded expansion and, according to Theorem 3.3, boundedstar chromatic number. It follows from Lemma 3.1 that ( Gaifman , Unfold ) is a transductionpairing of T and Gaifman ( T ). ◀ When we speak about transductions of permutations, we consider the permutations asdefined in Section 2.1. Hence the language used to define the transduction can use the binaryrelations < , < , as well as equality. Note that the mapping x σ ( x ) is not first-orderdefinable in this setting.Let σ ∈ S n . The permutation σ can be represented by the balanced ordered matching M <σ = ([2 n ] , E, < ), where < is the natural order on [2 n ] and E is a perfect matching inwhich i ∈ [ n ] is matched with σ ( i ) + n . ▷ Claim 7.1. Let c ∈ N and let C < be a class of ordered graphs with star chromatic numberat most c . There exist a copying-transduction T with bf( T ) = c , a simple transduction T ,and a class P of permutations such that ( T , T ) is a transduction pairing of C < and P . Proof. Let C be the reduct of C < obtained by “forgetting” the linear order. As C has starchromatic number at most c , each graph G ∈ C has an orientation with indegree at most c − c be the relational signature consisting of c − E , . . . , E c − and let Σ Unfold , Gaifman ) is a transduction pairing of C and D .This transduction also defines a transduction pairing ( T U , T G ) of C < and D < .The copying-transduction T is the composition of T B , a c -blowing, and the transductiondefined as follows: Let f i be the first-order definable function that maps a vertex x to itsin-neighbor in relation E i if it exists, and to x , otherwise. We further define f c as the identity.We define < as the lexicographic order and < as follows: ( x, i ) < ( y, j ) if f i ( x ) < f j ( y ) or f i ( x ) = f j ( y ) and i < j , or f i ( x ) = f j ( y ) , i = j , and x < y .The definition of T is as follows: we partition V into V , . . . , V c in such a way that themaximum element of < is in V c . The linear order < is the restriction of < to V c . Thenwe define the mapping g i : V c → V c as follows: for x ∈ V c , if there exists no x ′ ∈ V i suchthat x ′ < x and no element between x ′ and x (in < ) belongs to V i ∪ V c then g i ( x ) = x .Otherwise, g i ( x ) is, with respect to < , the minimum element of V c greater than x ′ .Let f , . . . , f c − be mappings on V and let < and < be defined by T . We let V i = V ×{ i } .By definition, the maximum element of V × [ c ] with respect to < is in V c . Let i ∈ [ c − x, c ) ∈ V c . By construction, x ′ = ( x, i ), and (in < ) the minimum element in V c greaterthan x ′ is ( f i ( x ) , c ). Thus g i (( x, c )) = ( f i ( x ) , c ). Hence T ◦ T subsumes the identity. Itfollows that ( T , T ) is a transduction pairing of ( C < ) and P = T ( C < ). ◁▶ Theorem 7.2. For every class C of binary structures with twin-width at most t there existsa proper class of permutations P , an integer k , and a transduction T from P to C , suchthat for every graph G ∈ C there is a permutation σ ∈ P on k | G | elements with G ∈ T ( σ ) . Proof. Let C be a class of binary structures with twin-width at most t . Let T be a class oftwin-models obtained by optimal contraction sequences of graphs in C , and let F be the classof the corresponding full twin-models. According to Lemma 5.2 F has twin-width at most 2 t ,moreover, applying the transduction L on ≺ we transform F into the class T < , whose reductis T . Let D < be the class obtained from T < be taking the Gaifman graphs of the relationsdistinct from the linear order, and keeping the linear order, and let D be the reduct of D < . Bonnet, J. Nešetřil, P. Ossona de Mendez, S. Siebertz, and S. Thomassé 15 obtained by forgetting the linear order. Thus D = Gaifman ( T ). As the classes D < andits reduct D are transductions of the class F they have bounded twin-width. Moreover,the class D is degenerate hence it has bounded expansion and, in particular, bounded starchromatic number. It follows that we have a transduction pairing of D < and a class P ofpermutations, which is proper as it has bounded twin-width. From the transduction pairingof T < and D < and the one of F and T < we deduce that there is a transduction pairingof F and P . As C is a transduction of F we conclude that C is a transduction of P . ◀ Note that any transduction from the class P of permutations to the class C of Σ-structuresobviously defines a transduction from the permutation class P obtained by closing P undersub-permutations to the class C . ▶ Corollary 7.3. Every class of graphs with bounded twin-width contains at most c n non-isomorphic graphs on n vertices (for some constant c depending on the class). The growth constant of a class C lab of labeled graphs is defined as γ lab C = lim sup( | C lab n | /n !) /n ,where C n denotes the set of all graphs in C which have n vertices. By analogy, the unlabeledgrowth constant of a class C of (unlabeled) graphs is defined as γ C = lim sup | C n | /n .Bounding the star chromatic number of a twin-model of a graph with twin-width d as afunction of d would allow to give some upper bound on the constant γ C for a class C withbounded twin-width.It has been conjectured in [4] that every small hereditary class of graphs has boundedtwin-width. According to Corollary 7.3 this is equivalent to the following conjecture. ▶ Conjecture 8.1. For a hereditary class of graphs C the following are equivalent: C has bounded twin-width; C contains at most c n non-isomorphic graphs with n vertices, for some constant c ; C is small, that is: C contains at most c n n ! labeled graphs with n vertices, for someconstant c . It was announced by Simon and Toruńczyk [19], and independently proved by Bonnet et al. [5] that a class of graphs C has bounded twin-width if and only if it is the reduct ofa monadically dependent class of ordered graphs. This implies the following duality typestatement for every class C < of ordered graphs: ∃ σ ( Av( σ ) (cid:47) (cid:47) (cid:47) (cid:47) C < ) ⇐⇒ C < / (cid:47) (cid:47) (cid:47) (cid:47) G , where Av( σ ) denotes the class of all permutations avoiding the pattern σ and G denote theclass of all graphs.The connection between classes of ordered graphs and permutation classes might well beeven deeper than what is proved in this paper. ▶ Conjecture 8.2. Every hereditary class C < of ordered graphs is transduction equivalent toa permutation class. This conjecture is known to hold in the following cases:if the class C < is not monadically dependent, as it is then transduction equivalent to theclass of all permutations [5];if the reduct C of C < is biclique-free, as either C < is not monadically dependent (previousitem), or it has bounded twin-width [5] thus, according to Theorem 3.2, Theorem 3.3and Claim 7.1 the class C < is transduction equivalent to a permutation class;if the reduct C of C < is a transduction of a class with bounded expansion as C isthen transduction equivalent to a bounded expansion class D [10] and this transductionequivalence can be extended to a transduction equivalence of C < and an expansion D < of D , which falls in the second item. References Michael Albert, Mathilde Bouvel, and Valentin Féray. Two first-order logics of permutations. Journal of Combinatorial Theory, Series A , 171:105158, 2020. Olivier Bernardi, Marc Noy, and Dominic Welsh. Growth constants of minor-closed classes ofgraphs. 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