Asymptotic growth of minor-closed classes of graphs Olivier - - PowerPoint PPT Presentation

Asymptotic growth of minor-closed classes of graphs

Olivier Bernardi - Centre de Recerca Matemàtica Joint work with Marc Noy and Dominic Welsh Humboldt-Universität zu Berlin, June 2007

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Growth of minor-closed classes

Graph minors. Rough classification (Polynomial, Exponential, Factorial) and refinements. Growth constants.

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Graph minors

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Deletion and contraction

Contraction Deletion

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Minors

A graph H is minor of a graph G if H can be obtained from G by a sequence of deletions and contractions. Example:

H ≺ G

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Minors

A graph H is minor of a graph G if H can be obtained from G by a sequence of deletions and contractions. Example:

H ≺ G G

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Minors

A graph H is minor of a graph G if H can be obtained from G by a sequence of deletions and contractions. Example:

H ≺ G H G

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Minor-closed classes

A class of graphs is a set of labeled graphs closed under isomorphism.

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Minor-closed classes

A class of graphs is a set of labeled graphs closed under isomorphism. A class of graphs is closed under minors if it is closed under deletions and contractions. Example: Planar graphs: Forests:

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Minor-closed classes

A class of graphs is a set of labeled graphs closed under isomorphism. A class of graphs is closed under minors if it is closed under deletions and contractions. A graph G is a minimal excluded minor for a class G if G is not in G but any proper minor of G is in G. Example: Planar graphs: Forests:

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Minor-closed classes

A class of graphs is a set of labeled graphs closed under isomorphism. A class of graphs is closed under minors if it is closed under deletions and contractions. A graph G is a minimal excluded minor for a class G if G is not in G but any proper minor of G is in G. Minor theorem [Robertson and Seymour]: For any minor-closed class, the number of minimal excluded minors is finite.

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Growth of minor-closed classes: a classification

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Upper-bound

gn : number of graphs with n vertices in G.

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Upper-bound

gn : number of graphs with n vertices in G. Theorem [Norine, Seymour, Thomas, Wallan]: For any proper minor-closed class G, there exists c such that gn ≤ cnn!.

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Classification

Theorem [Bernardi, Noy, Welsh]: The growth of a proper minor-closed class G, is either factorial n! ≤ gn ≤ cnn! for some c > 1, pseudo-factorial n!! ≤ gn < ǫnn! for all ǫ > 0, exponential 2n ≤ gn ≤ cn for some c > 2, polynomial

n(n−1) 2

≤ gn ≤ nc for some c ≥ 2,

• r constant

gn = 0 or 1.

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Classification : proof

The growth of a proper minor-closed class G, is either factorial n! ≤ gn ≤ cnn! for some c > 1, Path: pseudo-factorial n!! ≤ gn < ǫnn! for all ǫ > 0, matching: exponential 2n ≤ gn ≤ cn for some c > 2, star: polynomial

n(n−1) 2

≤ gn ≤ nc for some c ≥ 2, edge:

• r constant

gn = 0 or 1.

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Proof : path obstruction

Prop: If G contains all paths, then gn ≥ n!,

• therwise gn < ǫnn! for all ǫ > 0.

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Proof : path obstruction

Lemma: The class G = Ex(Pk) satisfies gn < ǫnn! for all ǫ > 0. Proof:

• The class ¯

G = Ex(Pk, ∆) satisfies ¯ gn < ǫnn! for all ǫ > 0.

< k

¯ G(z) ≤ F(z) = ezeze...zez

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Proof : path obstruction

Lemma: The class G = Ex(Pk) satisfies gn < ǫnn! for all ǫ > 0. Proof:

• The class ¯

G = Ex(Pk, ∆) satisfies ¯ gn < ǫnn! for all ǫ > 0.

< k

• The class G = Ex(Pk) satisfies gn < 2kn ¯

gn.

< k

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Proof : matching obstruction

Prop: If G contains all matchings, then gn ≥ (n − 1)!! = (n − 1)(n − 3) · · ·,

• therwise gn < cn for some c > 0.

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Proof : matching obstruction

Prop: If G contains all matchings, then gn ≥ (n − 1)!! = (n − 1)(n − 3) · · ·,

• therwise gn < cn for some c > 0.

Lemma: The class G = Ex(Mk) satisfies gn < P(n) · 22kn.

. . .

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Proof : star obstruction

Prop: If G contains all stars, then gn ≥ 2n,

• therwise gn < nc for some c > 0.

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Proof : star obstruction

Prop: If G contains all stars, then gn ≥ 2n,

• therwise gn < nc for some c > 0.

Lemma: The class G = Ex(Mk, Sk) satisfies gn < P(n).

. . .

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Proof : edge obstruction

Prop: If G contains the graphs with 1 edge, then gn ≥ n(n−1)

2

,

• therwise gn = 0 or 1 for large n.

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Refined classification

Theorem: The growth of a minor-closed class G, is either factorial n! ≤ gn ≤ cnn! for some c > 1, pseudo-factorial n!! ≤ gn < ǫnn! for all ǫ > 0, exponential 2n ≤ gn ≤ cn for some c > 2, polynomial

n(n−1) 2

≤ gn ≤ nc for some c ≥ 2,

• r constant

gn = 0 or 1.

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Refined classification

Theorem: The growth of a minor-closed class G, is either factorial n! ≤ gn ≤ cnn! for some c > 1, pseudo-factorial n!! ≤ gn < ǫnn! for all ǫ > 0, exponential 2n ≤ gn ≤ cn for some c > 2, polynomial

n(n−1) 2

≤ gn ≤ nc for some c ≥ 2,

• r constant

gn = 0 or 1. Theorem : If G has polynomial growth, then there is a polynomial P such that gn = P(n) for large n.

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Refined classification

Theorem: The growth of a minor-closed class G, is either factorial n! ≤ gn ≤ cnn! for some c > 1, pseudo-factorial n!! ≤ gn < ǫnn! for all ǫ > 0, exponential 2n ≤ gn ≤ cn for some c > 2, polynomial

n(n−1) 2

≤ gn ≤ nc for some c ≥ 2,

• r constant

gn = 0 or 1. Theorem : If G has exponential growth, then there is an integer k ≥ 2 such that gn ≍poly kn.

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Refined classification

Theorem: The growth of a minor-closed class G, is either factorial n! ≤ gn ≤ cnn! for some c > 1, pseudo-factorial n!! ≤ gn < ǫnn! for all ǫ > 0, exponential 2n ≤ gn ≤ cn for some c > 2, polynomial

n(n−1) 2

≤ gn ≤ nc for some c ≥ 2,

• r constant

gn = 0 or 1. Theorem : If G has pseudo-factorial growth, then either n(1−δ)n ≤ gn < ǫnnn for all δ, ǫ > 0,

• r there is an integer k such that gn ≍exp n

(k−1) k

n.

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Growth constants

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Growth constants

The growth constant of a class G is γ(G) = lim sup gn n! 1/n . We are interested in the set Γ = {γ(G) / G minor-closed}.

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Growth constants

The growth constant of a class G is γ(G) = lim sup gn n! 1/n . Example:

• G = Ex(Pk): growth constant 0.
• Path forests: growth constant 1.
• Caterpillar forests: growth constant ξ ≈1.76, root of e1/x =x.
• Forests: growth constant e ≈ 2.71.

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Class Growth Reference

Ex(Pk)

Path forests 1 Standard Caterpillars ξ ≈ 1.76 Forests = Ex(K3) e ≈ 2.71 Standard

Ex(C4)

3.63

Ex(K4 − e)

4.18

Ex(C5)

4.60 Outer-planar = Ex(K4, K2,3) 7.320 [Bodirsky et al.]

Ex(K2,3)

7.327 [Bodirsky et al.] Series parallel = Ex(K4) 9.07 [Bodirsky et al.]

Ex(W4)

11.54 [Gimenez et al.]

Ex(K5 − e)

12.96 [Gimenez et al.]

Ex(K2 × K3)

14.13 [Gimenez et al.] Planar 27.226 [Gimenez & Noy]

• Embed. in fixed surface

27.226 [McDiarmid]

Ex(K3,3)

27.229 [Gerke et al.]

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Apex construction (McDiarmid)

1

ξ e

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Apex construction (McDiarmid)

1

ξ e

2 2ξ 4 2e

Proposition: If γ is a growth constant, then 2γ also. Proof: γ(GA) = 2 · γ(G).

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Gaps

1

ξ e

2 2ξ 4 2e

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Gaps

1

ξ e

2 2ξ 4 2e

Proposition: There is no growth constant between 1 and ξ. Proof: Obstructions :

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Accumulation points

1

ξ e

2 2ξ 4 2e

Prop: The number e and the planar constant κ≈27.22 are accumulation points.

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Accumulation points

1

ξ e

2 2ξ 4 2e

Prop: The number e and the planar constant κ≈27.22 are accumulation points. Theorem: If G1, . . . , Gk are 2-connected and are not cycles, then γ = γ(Ex(G1, . . . , Gk)) is an accumulation point.

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Thanks.

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