Interpreting nowhere dense graph classes as a classical notion from model theory

Isolde Adler Goethe University Frankfurt

joint work with Hans Adler

AlMoTh, Universitat¨ Leipzig, 24.2.2011 Introduction

We show that fundamental notions from graph theory and infinite model theory coincide.

Theorem (A2 2010) For any graph class C that is closed under taking subgraphs, the following are equivalent: • C is nowhere dense, • C is stable, • C is not independent (NIP).

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 2/26 Outline

1. Nowhere dense graph classes 2. Stability 3. The independence property 4. Conclusion

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 3/26 Graphs are . . . undirected and simple.

V (G) := vertex set of G E(G) := set of edges e ⊆ V (G), |e| = 2 Graph H is a subgraph of G, if V (H) ⊆ V (G) and E(H) ⊆ E(G).

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 4/26 Graphs are . . . undirected and simple.

V (G) := vertex set of G E(G) := set of edges e ⊆ V (G), |e| = 2 Graph H is a subgraph of G, if V (H) ⊆ V (G) and E(H) ⊆ E(G).

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 4/26 Graphs are . . . undirected and simple.

V (G) := vertex set of G E(G) := set of edges e ⊆ V (G), |e| = 2 Graph H is a subgraph of G, if V (H) ⊆ V (G) and E(H) ⊆ E(G).

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 4/26 Graphs are . . . undirected and simple.

V (G) := vertex set of G E(G) := set of edges e ⊆ V (G), |e| = 2 Graph H is a subgraph of G, if V (H) ⊆ V (G) and E(H) ⊆ E(G).

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 4/26 Topological minors

Definition A subdivision of a graph H is a graph resulting from H by subdividing edges, i.e. by replacing edges by (new) paths. H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 5/26 Topological minors

Definition A subdivision of a graph H is a graph resulting from H by subdividing edges, i.e. by replacing edges by (new) paths. H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 5/26 Topological minors

Definition A subdivision of a graph H is a graph resulting from H by subdividing edges, i.e. by replacing edges by (new) paths. H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 5/26 Topological minors

Definition A subdivision of a graph H is a graph resulting from H by subdividing edges, i.e. by replacing edges by (new) paths. H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 5/26 Topological minors

Definition A subdivision of a graph H is a graph resulting from H by subdividing edges, i.e. by replacing edges by (new) paths. H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 5/26 Topological minors

Definition A subdivision of a graph H is a graph resulting from H by subdividing edges, i.e. by replacing edges by (new) paths. H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 5/26 Topological r-minors

Definition Let r ∈ N. An r-subdivision of a graph H is a graph resulting from H by subdividing edges at most r times H is a topological r-minor of G if an r-subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 6/26 Topological r-minors

Definition Let r ∈ N. An r-subdivision of a graph H is a graph resulting from H by subdividing edges at most r times H is a topological r-minor of G if an r-subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 6/26 Topological r-minors

Definition Let r ∈ N. An r-subdivision of a graph H is a graph resulting from H by subdividing edges at most r times H is a topological r-minor of G if an r-subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 6/26 Topological r-minors

Definition Let r ∈ N. An r-subdivision of a graph H is a graph resulting from H by subdividing edges at most r times H is a topological r-minor of G if an r-subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 6/26 Topological r-minors

Definition Let r ∈ N. An r-subdivision of a graph H is a graph resulting from H by subdividing edges at most r times H is a topological r-minor of G if an r-subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 6/26 Topological r-minors

Definition Let r ∈ N. An r-subdivision of a graph H is a graph resulting from H by subdividing edges at most r times H is a topological r-minor of G if an r-subdivision of H is isomorphic to a subgraph of G.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 6/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense graph classes

Kn := complete graph (clique) on n vertices

Definition (Nesetˇ rilˇ and de Mendez in 20081) Let C be a graph class. C is nowhere dense, if for every r ∈ N there is an n ∈ N such that no graph in C has Kn as a topological r-minor.

Examples • every finite graph class • acyclic graphs (r 7→ n := 3) • planar graphs (r 7→ n := 5) • graphs of degree ≤ d (r 7→ n := d + 2) • graphs excluding a fixed minor H (r 7→ n := |V (H)|)

• graphs locally excluding a minor Hr (r 7→ n := |V (Hr+1)|)

1in the context of homomorphism preservation theorems (nowhere dense = uniformly quasi-wide) ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 7/26 Nowhere dense = superflat r Let Kn := Kn, where every edge is subdivided exactly r times.

Definition (Podewski and Ziegler, 1978) r Class C is superflat, if for every r ∈ N there is an n ∈ N such that Kn is not a subgraph of any member of C.

Example

r • The class {Kr | 2 ≤ r ∈ N} is superflat.

• C is nowhere dense, if for every r there is an n such that no graph in C has Kn as a topological r-minor. • C is nowhere dense ⇐⇒ for every r there is an n such that no ≤r Kn is a subgraph of a member of C.

Remark C is superflat ⇐⇒ C is nowhere dense. Proof: Ramsey.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 8/26 Nowhere dense = superflat r Let Kn := Kn, where every edge is subdivided exactly r times.

Definition (Podewski and Ziegler, 1978) r Class C is superflat, if for every r ∈ N there is an n ∈ N such that Kn is not a subgraph of any member of C.

Example

r • The class {Kr | 2 ≤ r ∈ N} is superflat.

• C is nowhere dense, if for every r there is an n such that no graph in C has Kn as a topological r-minor. • C is nowhere dense ⇐⇒ for every r there is an n such that no ≤r Kn is a subgraph of a member of C.

Remark C is superflat ⇐⇒ C is nowhere dense. Proof: Ramsey.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 8/26 Nowhere dense = superflat r Let Kn := Kn, where every edge is subdivided exactly r times.

Definition (Podewski and Ziegler, 1978) r Class C is superflat, if for every r ∈ N there is an n ∈ N such that Kn is not a subgraph of any member of C.

Example

r • The class {Kr | 2 ≤ r ∈ N} is superflat.

• C is nowhere dense, if for every r there is an n such that no graph in C has Kn as a topological r-minor. • C is nowhere dense ⇐⇒ for every r there is an n such that no ≤r Kn is a subgraph of a member of C.

Remark C is superflat ⇐⇒ C is nowhere dense. Proof: Ramsey.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 8/26 Nowhere dense = superflat r Let Kn := Kn, where every edge is subdivided exactly r times.

Definition (Podewski and Ziegler, 1978) r Class C is superflat, if for every r ∈ N there is an n ∈ N such that Kn is not a subgraph of any member of C.

Example

r • The class {Kr | 2 ≤ r ∈ N} is superflat.

• C is nowhere dense, if for every r there is an n such that no graph in C has Kn as a topological r-minor. • C is nowhere dense ⇐⇒ for every r there is an n such that no ≤r Kn is a subgraph of a member of C.

Remark C is superflat ⇐⇒ C is nowhere dense. Proof: Ramsey.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 8/26 Nowhere dense = superflat r Let Kn := Kn, where every edge is subdivided exactly r times.

Definition (Podewski and Ziegler, 1978) r Class C is superflat, if for every r ∈ N there is an n ∈ N such that Kn is not a subgraph of any member of C.

Example

r • The class {Kr | 2 ≤ r ∈ N} is superflat.

• C is nowhere dense, if for every r there is an n such that no graph in C has Kn as a topological r-minor. • C is nowhere dense ⇐⇒ for every r there is an n such that no ≤r Kn is a subgraph of a member of C.

Remark C is superflat ⇐⇒ C is nowhere dense. Proof: Ramsey.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 8/26 Nowhere dense = superflat r Let Kn := Kn, where every edge is subdivided exactly r times.

Definition (Podewski and Ziegler, 1978) r Class C is superflat, if for every r ∈ N there is an n ∈ N such that Kn is not a subgraph of any member of C.

Example

r • The class {Kr | 2 ≤ r ∈ N} is superflat.

• C is nowhere dense, if for every r there is an n such that no graph in C has Kn as a topological r-minor. • C is nowhere dense ⇐⇒ for every r there is an n such that no ≤r Kn is a subgraph of a member of C.

Remark C is superflat ⇐⇒ C is nowhere dense. Proof: Ramsey.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 8/26 bounded clique-width

bounded local clique-width

Graph classes

paths

bounded degree bounded tree-width planar

bounded local tree-width excluded minor

locally excluded minor bounded expansion

nowhere dense

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 9/26 Graph classes

paths

bounded degree bounded tree-width planar

bounded clique-width bounded local tree-width excluded minor

bounded local clique-width locally excluded minor bounded expansion

nowhere dense

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 9/26 Outline

1. Nowhere dense graph classes 2. Stability 3. The independence property 4. Conclusion

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 10/26 Logic

• first-order logic • relational structures • we code undirected graphs as {E}-structures, where E is a symmetric, irreflexive binary relation

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 11/26 Stability

Let C be a class of structures of a fixed signature.

Definition A first-order formula ϕ(x¯, y¯) has the order property on C if for every n ¯ ¯ there exist a structure M ∈ C and tuples a¯1,..., a¯n, b1,..., bn ∈ M such that ¯ M |= ϕ(a¯i , bj ) ⇐⇒ i < j.

A class C of structures is called stable if there is no such formula.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 12/26 Stability

Let C be a class of structures of a fixed signature.

Definition A first-order formula ϕ(x¯, y¯) has the order property on C if for every n ¯ ¯ there exist a structure M ∈ C and tuples a¯1,..., a¯n, b1,..., bn ∈ M such that ¯ M |= ϕ(a¯i , bj ) ⇐⇒ i < j.

A class C of structures is called stable if there is no such formula.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 12/26 Stability

Let C be a class of structures of a fixed signature.

Definition A first-order formula ϕ(x¯, y¯) has the order property on C if for every n ¯ ¯ there exist a structure M ∈ C and tuples a¯1,..., a¯n, b1,..., bn ∈ M such that ¯ M |= ϕ(a¯i , bj ) ⇐⇒ i < j.

A class C of structures is called stable if there is no such formula.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 12/26 Formula ϕ has the order property ¯ a¯1 ϕ b1

¯ a¯2 b2

¯ a¯3 b3

¯ a¯4 b4

¯ a¯5 b5

¯ a¯6 b6

¯ a¯7 b7 ¯ M |= ϕ(a¯i , bj ) ⇐⇒ i < j

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 13/26 Example The class of graphs Bn (where B7 is shown below) is not stable, witnessed by E(x, y).

a1 b1

a2 b2

a3 b3

a4 b4

a5 b5

a6 b6

a7 b7

B7 |= E(ai , bj ) ⇐⇒ i < j

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 14/26 Podewski-Ziegler for graph classes

Theorem (Podewski, Ziegler 1978) Let G be an infinite graph (coded as an {E}-structure). If {G} is superflat2 then {G} is stable.

Lemma (Podewski-Ziegler for graph classes) Let C be a graph class. If C is superflat then C is stable.

Proof sketch.

• Encode C in a single graph GC s.t. C superflat ⇒ {GC} superflat

• interpret C in {GC}

• apply Podewski-Ziegler-Theorem to {GC}

2 r C is superflat, if for every r there is an n such that Kn is not a subgraph of any member of C. ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 15/26 Podewski-Ziegler for graph classes

Theorem (Podewski, Ziegler 1978) Let G be an infinite graph (coded as an {E}-structure). If {G} is superflat2 then {G} is stable.

Lemma (Podewski-Ziegler for graph classes) Let C be a graph class. If C is superflat then C is stable.

Proof sketch.

• Encode C in a single graph GC s.t. C superflat ⇒ {GC} superflat

• interpret C in {GC}

• apply Podewski-Ziegler-Theorem to {GC}

2 r C is superflat, if for every r there is an n such that Kn is not a subgraph of any member of C. ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 15/26 Podewski-Ziegler for graph classes

Theorem (Podewski, Ziegler 1978) Let G be an infinite graph (coded as an {E}-structure). If {G} is superflat2 then {G} is stable.

Lemma (Podewski-Ziegler for graph classes) Let C be a graph class. If C is superflat then C is stable.

Proof sketch.

• Encode C in a single graph GC s.t. C superflat ⇒ {GC} superflat

• interpret C in {GC}

• apply Podewski-Ziegler-Theorem to {GC}

2 r C is superflat, if for every r there is an n such that Kn is not a subgraph of any member of C. ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 15/26 Podewski-Ziegler for graph classes

Theorem (Podewski, Ziegler 1978) Let G be an infinite graph (coded as an {E}-structure). If {G} is superflat2 then {G} is stable.

Lemma (Podewski-Ziegler for graph classes) Let C be a graph class. If C is superflat then C is stable.

Proof sketch.

• Encode C in a single graph GC s.t. C superflat ⇒ {GC} superflat

• interpret C in {GC}

• apply Podewski-Ziegler-Theorem to {GC}

2 r C is superflat, if for every r there is an n such that Kn is not a subgraph of any member of C. ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 15/26 Podewski-Ziegler for graph classes

Theorem (Podewski, Ziegler 1978) Let G be an infinite graph (coded as an {E}-structure). If {G} is superflat2 then {G} is stable.

Lemma (Podewski-Ziegler for graph classes) Let C be a graph class. If C is superflat then C is stable.

Proof sketch.

• Encode C in a single graph GC s.t. C superflat ⇒ {GC} superflat

• interpret C in {GC}

• apply Podewski-Ziegler-Theorem to {GC}

2 r C is superflat, if for every r there is an n such that Kn is not a subgraph of any member of C. ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 15/26 Podewski-Ziegler for graph classes

Theorem (Podewski, Ziegler 1978) Let G be an infinite graph (coded as an {E}-structure). If {G} is superflat2 then {G} is stable.

Lemma (Podewski-Ziegler for graph classes) Let C be a graph class. If C is superflat then C is stable.

Proof sketch.

• Encode C in a single graph GC s.t. C superflat ⇒ {GC} superflat

• interpret C in {GC}

• apply Podewski-Ziegler-Theorem to {GC}

2 r C is superflat, if for every r there is an n such that Kn is not a subgraph of any member of C. ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 15/26 Outline

1. Nowhere dense graph classes 2. Stability 3. The independence property 4. Conclusion

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 16/26 The independence property

Let C be a class of structures of a fixed signature.

Definition A first-order formula ϕ(x¯, y¯) has the independence property on C if for every n there is a structure M ∈ C and tuples a¯1,..., a¯n ∈ M and ¯ bJ ∈ M for all J ⊆ {1,..., n} such that ¯ M |= ϕ(a¯i , bJ ) ⇐⇒ i ∈ J.

C is NIP (‘Not the Independence Property’) if no formula has the independence property on C.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 17/26 The independence property

Let C be a class of structures of a fixed signature.

Definition A first-order formula ϕ(x¯, y¯) has the independence property on C if for every n there is a structure M ∈ C and tuples a¯1,..., a¯n ∈ M and ¯ bJ ∈ M for all J ⊆ {1,..., n} such that ¯ M |= ϕ(a¯i , bJ ) ⇐⇒ i ∈ J.

C is NIP (‘Not the Independence Property’) if no formula has the independence property on C.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 17/26 The independence property

Let C be a class of structures of a fixed signature.

Definition A first-order formula ϕ(x¯, y¯) has the independence property on C if for every n there is a structure M ∈ C and tuples a¯1,..., a¯n ∈ M and ¯ bJ ∈ M for all J ⊆ {1,..., n} such that ¯ M |= ϕ(a¯i , bJ ) ⇐⇒ i ∈ J.

C is NIP (‘Not the Independence Property’) if no formula has the independence property on C.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 17/26 Formula ϕ has the independence property

a¯1 ϕ

a¯2

a¯3

¯ a¯4 b{2,4,5}

a¯5

a¯6

¯ M |= ϕ(a¯i , bJ ) ⇐⇒ i ∈ J

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 18/26 Stability & NIP

Remark If C is stable then C is NIP.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 19/26 C not NIP ⇒ C not stable

¯ a¯1 ϕ b1 a¯1 ϕ ¯ a¯2 b2 a¯2 ¯ a¯3 b3 a¯3 ¯ ¯ ¯ a4 b4 a¯4 b{2,4,5} ¯ a¯5 b5 a¯5 ¯ a¯6 b6 a¯6 ¯ a¯7 b7

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 20/26 Main Theorem

Theorem C a graph class closed under taking subgraphs. The following are equivalent. 1. C is nowhere dense. 2. C is superflat. 3. C is stable. 4. C is NIP.

Remark: closure under subgraphs only for ‘4 ⇒ 1’.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 21/26 Main Theorem

Theorem C a graph class closed under taking subgraphs. The following are equivalent. 1. C is nowhere dense. 2. C is superflat. 3. C is stable. 4. C is NIP.

Remark: closure under subgraphs only for ‘4 ⇒ 1’.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 21/26 paths

bounded degree bounded tree-width planar

bounded clique-width bounded local tree-width excluded minor

bounded local clique-width locally excluded minor bounded expansion

nowhere dense

stable

NIP

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 22/26 paths

bounded degree bounded tree-width planar

bounded clique-width bounded local tree-width excluded minor

bounded local clique-width locally excluded minor bounded expansion

nowhere dense

stable

NIP

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 22/26 Outline

1. Nowhere dense graph classes 2. Stability 3. The independence property 4. Conclusion

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 23/26 Conclusion: Outlook

Theorem C a class of structures over a fixed finite signature, C closed under subgraphs. The following are equivalent. 1. C is nowhere dense. 2. C is superflat. 3. C is stable. 4. C is stable. 5. C is NIP. 6. C is NIP.

• Computational learning theory: C subgraph-closed. First order definable concepts are PAC learnable on C ⇐⇒ C is nowhere dense.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 24/26 Conclusion: Outlook

Theorem C a class of structures over a fixed finite signature, C closed under subgraphs. The following are equivalent. 1. C is nowhere dense. 2. C is superflat. 3. C is stable. 4. C is stable. 5. C is NIP. 6. C is NIP.

• Computational learning theory: C subgraph-closed. First order definable concepts are PAC learnable on C ⇐⇒ C is nowhere dense.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 24/26 Open Problems

• Is there a simple structural characterisation for NIP graph classes (that are closed under taking induced subgraphs)? • Is first order model checking in FPT on nowhere dense graph classes? • Explore connections between infinite model theory and algorithmic graph theory

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 25/26 paths

bounded degree bounded tree-width planar

bounded clique-width bounded local tree-width excluded minor

bounded local clique-width locally excluded minor bounded expansion

nowhere dense

stable Vielen Dank!

NIP

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 26/26 Why ‘nowhere dense’?

Let C be a graph class. C∇r := class of all topological r-minors of graphs in C.

Then: C∇0 = { all subgraphs of graphs in C}. Theorem (Nesetˇ ril,ˇ de Mendez, 2008) C a class of finite graphs. Then

log |E(H)| lim lim sup ∈ {0, 1, 2}. r→∞ H∈C∇r log |V (H)| |V (H)|→∞

Moreover, the quadratic case (right-hand side 2) is equivalent to: for some r there is no finite upper bound on the sizes of cliques that occur as topological r-minors. Original definition: nowhere dense:= right hand side ≤ 1.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 27/26 Why ‘nowhere dense’?

Let C be a graph class. C∇r := class of all topological r-minors of graphs in C.

Then: C∇0 = { all subgraphs of graphs in C}. Theorem (Nesetˇ ril,ˇ de Mendez, 2008) C a class of finite graphs. Then

log |E(H)| lim lim sup ∈ {0, 1, 2}. r→∞ H∈C∇r log |V (H)| |V (H)|→∞

Moreover, the quadratic case (right-hand side 2) is equivalent to: for some r there is no finite upper bound on the sizes of cliques that occur as topological r-minors. Original definition: nowhere dense:= right hand side ≤ 1.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 27/26 Why ‘nowhere dense’?

Let C be a graph class. C∇r := class of all topological r-minors of graphs in C.

Then: C∇0 = { all subgraphs of graphs in C}. Theorem (Nesetˇ ril,ˇ de Mendez, 2008) C a class of finite graphs. Then

log |E(H)| lim lim sup ∈ {0, 1, 2}. r→∞ H∈C∇r log |V (H)| |V (H)|→∞

Moreover, the quadratic case (right-hand side 2) is equivalent to: for some r there is no finite upper bound on the sizes of cliques that occur as topological r-minors. Original definition: nowhere dense:= right hand side ≤ 1.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 27/26 Why ‘nowhere dense’?

Let C be a graph class. C∇r := class of all topological r-minors of graphs in C.

Then: C∇0 = { all subgraphs of graphs in C}. Theorem (Nesetˇ ril,ˇ de Mendez, 2008) C a class of finite graphs. Then

log |E(H)| lim lim sup ∈ {0, 1, 2}. r→∞ H∈C∇r log |V (H)| |V (H)|→∞

Moreover, the quadratic case (right-hand side 2) is equivalent to: for some r there is no finite upper bound on the sizes of cliques that occur as topological r-minors. Original definition: nowhere dense:= right hand side ≤ 1.

ISOLDE ADLER NOWHEREDENSEGRAPHCLASSES, STABILITY, ANDINDEPENDENCE 27/26