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Background Enumeration Largest block 3-connected component Graph classes with given 3 -connected components: asymptotic enumeration and random graphs Juanjo Ru e (joint work with Omer Gim enez and Marc Noy) Laboratoire dInformatique

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Background Enumeration Largest block 3-connected component

Graph classes with given 3-connected components: asymptotic enumeration and random graphs

Juanjo Ru´ e

(joint work with Omer Gim´ enez and Marc Noy)

Laboratoire d’Informatique, ´ Ecole Polytechnique,

Workshop on Random Graphs and Maps on Surfaces, IHP, 2nd November 2009

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Background Enumeration Largest block 3-connected component

Background and definitions

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Background Enumeration Largest block 3-connected component

Objects: graphs

Labelled Graph= vertices+edges

1 3 2 4 5

Simple Graph= NO multiples edges, NO loops

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Background Enumeration Largest block 3-connected component

Objects: graphs

crucial concept: graph minor; tools: connectivy K4 is a minor of this graph connected→2-connected 2-connected→3-connected

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Background Enumeration Largest block 3-connected component

Generating functions

For a combinatorial class A we construct an ordinary generating function (OGF) A(x) such that A(x) =

∞

  • n=0

anxn. For labelled families of graphs we use exponential GFs (EGF): A(x, y) =

∞

  • n=0

an,m xn n! ym, where x marks vertices and y marks edges.

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Background Enumeration Largest block 3-connected component

The Symbolic Method

COMBINATORIAL RELATIONS between CLASSES

  • EQUATIONS between GENERATING FUNCTIONS

Construction OGF EGF Union A ∪ B A(x) + B(x) A(x) + B(x) Product A × B A(x) · B(x) − Labelled Product A ∗ B − A(x) · B(x) Sequence Seq (A)

1 1−A(x) 1 1−A(x)

Set Set (A) − exp (A(x)) Pointing A• x ∂

∂xA(x)

x ∂

∂xA(x)

Substitution A ◦ B A (B(x)) A (B(x))

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Background Enumeration Largest block 3-connected component

Singularity analysis on generating functions

GFs: analytic functions in a neighbourhood of the origin. The smallest singularity of A(z) determines the asymptotics

  • f the coefficients of A(z).
  • POSITION: exponential growth ρ.
  • NATURE: subexponetial growth.
  • Transfer Theorems: Let α /

∈ {0, −1, −2, . . .}. If A(z) = a · (1 − z/ρ)−α + O((1 − z/ρ)−α) then [zn]A(z) = a Γ(α) · nα−1 · ρ−n(1 + O(1))

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Background Enumeration Largest block 3-connected component

Singularity analysis on generating functions

GFs: analytic functions in a neighbourhood of the origin. The smallest singularity of A(z) determines the asymptotics

  • f the coefficients of A(z).
  • POSITION: exponential growth ρ.
  • NATURE: subexponetial growth.
  • Transfer Theorems: Let α /

∈ {0, −1, −2, . . .}. If A(z) = a · (1 − z/ρ)−α + O((1 − z/ρ)−α) then [zn]A(z) = a Γ(α) · nα−1 · ρ−n(1 + O(1))

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Background Enumeration Largest block 3-connected component

Singularity analysis on generating functions

GFs: analytic functions in a neighbourhood of the origin. The smallest singularity of A(z) determines the asymptotics

  • f the coefficients of A(z).
  • POSITION: exponential growth ρ.
  • NATURE: subexponetial growth.
  • Transfer Theorems: Let α /

∈ {0, −1, −2, . . .}. If A(z) = a · (1 − z/ρ)−α + O((1 − z/ρ)−α) then [zn]A(z) = a Γ(α) · nα−1 · ρ−n(1 + O(1))

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Background Enumeration Largest block 3-connected component

Limit laws

Study of parameters→ A(u, z) = ∞

n,m=0 an,mznum.

For a fixed n, the numbers an,m describes a discrete probability law Xn p(Xn = m) = an,m ∞

m=0 an,m

= [umzn]A(u, z) [zn]A(1, z) Does Xn converge in distribution to a random variable X? We expect normal limit distributions: general theorems We deal with parameters where these results do not apply.

  • Composition Schemes
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Background Enumeration Largest block 3-connected component

Limit laws

Study of parameters→ A(u, z) = ∞

n,m=0 an,mznum.

For a fixed n, the numbers an,m describes a discrete probability law Xn p(Xn = m) = an,m ∞

m=0 an,m

= [umzn]A(u, z) [zn]A(1, z) Does Xn converge in distribution to a random variable X? We expect normal limit distributions: general theorems We deal with parameters where these results do not apply.

  • Composition Schemes
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Background Enumeration Largest block 3-connected component

Definitions, enumeration and normal limit laws

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Background Enumeration Largest block 3-connected component

Our starting point

Asymptotic enumeration and limit laws of planar graphs (Gim´ enez, Noy) g1 · n−7/2 · γn

1 · n!

Asymptotic enumeration and limit laws of series-parallel graphs (Bodirsky, Gim´ enez, Kang, Noy) g2 · n−5/2 · γn

2 · n!

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Background Enumeration Largest block 3-connected component

Our starting point

Asymptotic enumeration and limit laws of planar graphs [Gim´ enez, Noy] g1 · n−7/2 · γn

1 · n!

Asymptotic enumeration and limit laws of series-parallel graphs [Bodirsky, Gim´ enez, Kang, Noy] g2 · n−5/2 · γn

2 · n!

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Background Enumeration Largest block 3-connected component

Our starting point

g1 · n−7/2 · γn

1 · n!

g2 · n−5/2 · γn

2 · n!

  • THE SUBEXPONENTIAL TERM GIVES THE

“PHYSICS” OF THE GRAPHS

  • GENERAL FRAMEWORK TO UNDERSTAND THIS

EXPONENT

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Background Enumeration Largest block 3-connected component

Our starting point

g1 · n−7/2 · γn

1 · n!

g2 · n−5/2 · γn

2 · n!

  • THE SUBEXPONENTIAL TERM GIVES THE

“PHYSICS” OF THE GRAPHS

  • GENERAL FRAMEWORK TO UNDERSTAND THIS

EXPONENT

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Background Enumeration Largest block 3-connected component

The strategy

networks Planar graphs: 3 − connected planar graphs 3 − conn. 2 − conn. conn. GENERAL

We get the equations in the opposite way

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Background Enumeration Largest block 3-connected component

The strategy

networks 3 − conn. 2 − conn. conn. GENERAL Series-parallel graphs: −

We get the equations in the opposite way

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Background Enumeration Largest block 3-connected component

The strategy

networks 3 − conn. 2 − conn. conn. GENERAL A family of 3-connected graphs

We get the equations in the opposite way

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Background Enumeration Largest block 3-connected component

General graphs from connected graphs

Let C be a family of connected graphs. G : graphs such that their connected components are in C. G = Set(C) = ⇒ G(x, y) = exp(C(x, y))

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Background Enumeration Largest block 3-connected component

General graphs from connected graphs

Let C be a family of connected graphs. G : graphs such that their connected components are in C. G = Set(C) = ⇒ G(x, y) = exp(C(x, y))

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Background Enumeration Largest block 3-connected component

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs. C : connected graphs with blocks in B. In other words, a vertex-rooted connected graph is a tree of 2-connected blocks.

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Background Enumeration Largest block 3-connected component

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs. C : connected graphs with blocks in B. In other words, a vertex-rooted connected graph is a tree of 2-connected blocks.

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Background Enumeration Largest block 3-connected component

Connected graphs from 2-connected graphs

A vertex-rooted connected graph is a tree of rooted blocks. C• = v × Set(B◦(v ← C•)) = ⇒ xC′(x, y) = x exp B′(xC′(x, y), y)

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Background Enumeration Largest block 3-connected component

2-connected graphs from 3-connected graphs

Decomposition in 3-connected components is slightly harder. Let T be a family of 3-connected graphs: T(x, z). We define B as those 2-connected graphs such that can be

  • btained from series, parallel, and T -compositions.

D(x, y) = (1 + y) exp xD2 1 + xD + 1 2x2 ∂T ∂z (x, D)

  • − 1

2(1 + y)∂B ∂y (x, y) = x2 (1 + D(x, y)) D is the GF for networks (essentially edge-rooted 2-connected graphs without the root and distinguished end-vertices).

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Background Enumeration Largest block 3-connected component

2-connected graphs from 3-connected graphs

Decomposition in 3-connected components is slightly harder. Let T be a family of 3-connected graphs: T(x, z). We define B as those 2-connected graphs such that can be

  • btained from series, parallel, and T -compositions.

D(x, y) = (1 + y) exp xD2 1 + xD + 1 2x2 ∂T ∂z (x, D)

  • − 1

2(1 + y)∂B ∂y (x, y) = x2 (1 + D(x, y)) D is the GF for networks (essentially edge-rooted 2-connected graphs without the root and distinguished end-vertices).

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Background Enumeration Largest block 3-connected component

A set of equations

INPUT: T(x, z)        1 2x2D ∂T ∂z (x, D) − log 1 + D 1 + y

  • +

xD2 1 + xD = 0 ∂B ∂y (x, y) = x2 2 1 + D(x, y) 1 + y

  C•(x, y) = exp (B◦(C•(x, y), y)) G(x, y) = exp(C(x, y))

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Background Enumeration Largest block 3-connected component

Examples of families & excluded minors (I)

  • Series-parallel graphs
  • Excluded minors:
  • T : None.
  • T(x, z) = 0.
  • Planar graphs
  • Excluded minors:
  • T : 3-connected planar graphs.
  • T(x, z): The number of labelled 2-connected planar graphs

(Bender, Gao, Wormald, 2002)

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Background Enumeration Largest block 3-connected component

Examples of families & excluded minors (II)

  • W4-free
  • Excluded minors:
  • T :
  • T(x, z) = 1

4!x4z6.

  • K−

5 -free

  • Excluded minors:
  • T :

, . . .

  • T(x, z) = 70

6! x6z9 − 1 2x

  • log(1 − xz2) + 2xz2 + x2z4

.

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Background Enumeration Largest block 3-connected component

Examples of families & excluded minors (III)

  • K3,3-free (Gerke, Gim´

enez, Noy, Weibl, 2006)

  • Excluded minors:
  • 3-connected components:

, 3-connected planar graphs.

  • T(x, z) = . . . .
  • If G = Ex(M) and all the excluded minors M are

3-connected, then G can be expressed in terms of its 3-connected graphs.

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Background Enumeration Largest block 3-connected component

RESULT: asymptotic enumeration

If either ∂T

∂z (x, z)

  • has no singularity, or
  • the singularity type is (1 − z/z0)α with α < 1,

then the situation is alike to the series-parallel case: D(x) ∼ d · (1 − x/x0)1/2 B(x) ∼ b · (1 − x/x0)3/2 C(x) ∼ c · (1 − x/ρ)3/2 G(x) ∼ g · (1 − x/ρ)3/2 dn ∼ d · n−3/2 · x−n · n! bn ∼ b · n−5/2 · x−n · n! cn ∼ c · n−5/2 · ρ−n · n! gn ∼ g · n−5/2 · ρ−n · n!

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Background Enumeration Largest block 3-connected component

RESULT: asymptotic enumeration (II)

If ∂T

∂z (x, z) has singularity type (1 − z/z0)3/2, then 3 different

situations may happen. Case 1 (Planar case) D(x) ∼ d · (1 − x/x0)3/2 B(x) ∼ b · (1 − x/x0)5/2 C(x) ∼ c · (1 − x/ρ)5/2 G(x) ∼ g · (1 − x/ρ)5/2 dn ∼ d · n−5/2 · x−n · n! bn ∼ b · n−7/2 · x−n · n! cn ∼ c · n−7/2 · ρ−n · n! gn ∼ g · n−7/2 · ρ−n · n!

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Background Enumeration Largest block 3-connected component

RESULT: asymptotic enumeration (II)

If ∂T

∂z (x, z) has singularity type (1 − z/z0)3/2, then 3 different

situations may happen. Case 2 (Series-parallel case) D(x) ∼ d · (1 − x/x0)1/2 B(x) ∼ b · (1 − x/x0)3/2 C(x) ∼ c · (1 − x/ρ)3/2 G(x) ∼ g · (1 − x/ρ)3/2 dn ∼ d · n−3/2 · x−n · n! bn ∼ b · n−5/2 · x−n · n! cn ∼ c · n−5/2 · ρ−n · n! gn ∼ g · n−5/2 · ρ−n · n!

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Background Enumeration Largest block 3-connected component

RESULT: asymptotic enumeration (II)

If ∂T

∂z (x, z) has singularity type (1 − z/z0)3/2, then 3 different

situations may happen. Case 3 (Mixed case) D(x) ∼ d · (1 − x/x0)3/2 B(x) ∼ b · (1 − x/x0)5/2 C(x) ∼ c · (1 − x/ρ)3/2 G(x) ∼ g · (1 − x/ρ)3/2 dn ∼ d · n−5/2 · x−n · n! bn ∼ b · n−7/2 · x−n · n! cn ∼ c · n−5/2 · ρ−n · n! gn ∼ g · n−5/2 · ρ−n · n!

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Background Enumeration Largest block 3-connected component

RESULT: asymptotic enumeration (III)

If ∂T

∂z (x, z) has singularity type (1 − z/z0)3/2, and a critical

condition happen, 2 cases can occur. Case 1 (Critical case 1) D(x) ∼ d · (1 − x/x0)5/3 B(x) ∼ b · (1 − x/x0)5/3 C(x) ∼ c · (1 − x/ρ)3/2 G(x) ∼ g · (1 − x/ρ)3/2 dn ∼ d · n−8/3 · x−n · n! bn ∼ b · n−8/3 · x−n · n! cn ∼ c · n−5/2 · ρ−n · n! gn ∼ g · n−5/2 · ρ−n · n!

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Background Enumeration Largest block 3-connected component

RESULT: asymptotic enumeration (III)

If ∂T

∂z (x, z) has singularity type (1 − z/z0)3/2, and a critical

condition happen, 2 cases can occur. Case 1 (Critical case 2) D(x) ∼ d · (1 − x/x0)5/2 B(x) ∼ b · (1 − x/x0)5/2 C(x) ∼ c · (1 − x/ρ)5/3 G(x) ∼ g · (1 − x/ρ)5/3 dn ∼ d · n−7/2 · x−n · n! bn ∼ b · n−7/2 · x−n · n! cn ∼ c · n−8/3 · ρ−n · n! gn ∼ g · n−8/3 · ρ−n · n!

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Background Enumeration Largest block 3-connected component

A critical phenomenon

The previous result must be understood for a fixed value of y There is a “bijection” between y and the density of edges. −yρ′(y)/ρ(y) = λ

  • Values of y = 1 give the subfamily of graphs with a prescribed

density of edges λ(y)n + O(√n).

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Background Enumeration Largest block 3-connected component

A critical phenomenon

The previous result must be understood for a fixed value of y There is a “bijection” between y and the density of edges. −yρ′(y)/ρ(y) = λ

  • Values of y = 1 give the subfamily of graphs with a prescribed

density of edges λ(y)n + O(√n).

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Background Enumeration Largest block 3-connected component

A critical phenomenon

In planar graphs, the qualitative behaviour is always the same, but we can construct families with different behaviour EXAMPLE: T = {3-connected cubic planar graphs}. b(λ)n: 2-connected graphs of G with (λn − √n, λn + √n) edges.

  • If λ < 1,31725 (y = 0,074223),

b(λ)n ∼ bλ · n−7/2 · x0(λ)−n · n!

  • If λ > 1,31725 (y = 0,074223),

b(λ)n ∼ bλ · n−5/2 · x0(λ)−n · n! Something similar is true for connected graphs: λ = 1,18441, y = 0,06705

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Background Enumeration Largest block 3-connected component

A critical phenomenon

In planar graphs, the qualitative behaviour is always the same, but we can construct families with different behaviour EXAMPLE: T = {3-connected cubic planar graphs}. b(λ)n: 2-connected graphs of G with (λn − √n, λn + √n) edges.

  • If λ < 1,31725 (y = 0,074223),

b(λ)n ∼ bλ · n−7/2 · x0(λ)−n · n!

  • If λ > 1,31725 (y = 0,074223),

b(λ)n ∼ bλ · n−5/2 · x0(λ)−n · n! Something similar is true for connected graphs: λ = 1,18441, y = 0,06705

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Background Enumeration Largest block 3-connected component

A critical phenomenon

In planar graphs, the qualitative behaviour is always the same, but we can construct families with different behaviour EXAMPLE: T = {3-connected cubic planar graphs}. b(λ)n: 2-connected graphs of G with (λn − √n, λn + √n) edges.

  • If λ < 1,31725 (y = 0,074223),

b(λ)n ∼ bλ · n−7/2 · x0(λ)−n · n!

  • If λ > 1,31725 (y = 0,074223),

b(λ)n ∼ bλ · n−5/2 · x0(λ)−n · n! Something similar is true for connected graphs: λ = 1,18441, y = 0,06705

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Background Enumeration Largest block 3-connected component

RESULT: limit laws

Many parameters follow either a normal limit law or a discrete limit law Normal: number of edges, number of blocks, number of cut vertices, number of copies of a given subgraph. Discrete: number of connected component, size of the largest connected component. SAME BEHAVIOUR IN ALL FAMILIES

  • ARE THERE PARAMETERS WITH DIFFERENT

BEHAVIOUR?

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Background Enumeration Largest block 3-connected component

RESULT: limit laws

Many parameters follow either a normal limit law or a discrete limit law Normal: number of edges, number of blocks, number of cut vertices, number of copies of a given subgraph. Discrete: number of connected component, size of the largest connected component. SAME BEHAVIOUR IN ALL FAMILIES

  • ARE THERE PARAMETERS WITH DIFFERENT

BEHAVIOUR?

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Background Enumeration Largest block 3-connected component

The largest block

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The Airy distribution

Let Ai(x) be the solution

1 2π

∞

−∞ exp

  • i
  • xt + t3/3
  • dt of the

differential equation y′′ − xy = 0. g(x) = 2e−2x3/3(xAi(x2) − Ai′(x2))

K 4 K 3 K 2 K 1 1 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7

left tail decays polynomially, right tail decays exponentially.

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Background Enumeration Largest block 3-connected component

Core in general maps

The block which contains the root is the core of the map.

M(z, u) = C(uz(1 + M(z))2)

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Background Enumeration Largest block 3-connected component

Core in general maps

The block which contains the root is the core of the map.

M(z, u) = C(uz(1 + M(z))2)

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Background Enumeration Largest block 3-connected component

Core in general maps

The block which contains the root is the core of the map.

M(z, u) = C(uz(1 + M(z))2)

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Background Enumeration Largest block 3-connected component

Core in general maps

The block which contains the root is the core of the map.

M(z, u) = C(uz(1 + M(z))2)

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Background Enumeration Largest block 3-connected component

The size of the core

Question: Let Xn be the size (in edges) of the largest block of a random map with n vertices (uniformly distributed). Which is its distribution?

  • Gao, Wormald (1999): the largest block has almost

surely n/3 edges, with deviations of order n2/3.

  • Banderier, Flajolet, Schaeffer, Soria (2001): critical

schemes 3

2 ◦ 3 2 obey an Airy law. (Universal property!)

C(uz(1 + M(z))2 = C(uH(z))→ C(z) = c − c1(1 − z/σ) + c3/2(1 − z/σ)3/2 + . . . H(z) = σ − h1(1 − z/ρ) + h3/2(1 − z/ρ)3/2 + . . .

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Background Enumeration Largest block 3-connected component

The size of the core

Question: Let Xn be the size (in edges) of the largest block of a random map with n vertices (uniformly distributed). Which is its distribution?

  • Gao, Wormald (1999): the largest block has almost

surely n/3 edges, with deviations of order n2/3.

  • Banderier, Flajolet, Schaeffer, Soria (2001): critical

schemes 3

2 ◦ 3 2 obey an Airy law. (Universal property!)

C(uz(1 + M(z))2 = C(uH(z))→ C(z) = c − c1(1 − z/σ) + c3/2(1 − z/σ)3/2 + . . . H(z) = σ − h1(1 − z/ρ) + h3/2(1 − z/ρ)3/2 + . . .

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Background Enumeration Largest block 3-connected component

2-connected core in connected graphs

We look for the root vertex of the graph:

  • If the root is NOT a cut vertex: the core is the unique

block which contains the root.

  • If the root is a cut vertex: the graph is CORELESS.

B′(uxC′(x))+

  • Graphs with core

exp

  • B′(xC′(x))
  • − B′(xC′(x))
  • Coreless graphs
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Background Enumeration Largest block 3-connected component

2-connected core in connected graphs

We look for the root vertex of the graph:

  • If the root is NOT a cut vertex: the core is the unique

block which contains the root.

  • If the root is a cut vertex: the graph is CORELESS.

B′(uxC′(x))+

  • Graphs with core

exp

  • B′(xC′(x))
  • − B′(xC′(x))
  • Coreless graphs
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Background Enumeration Largest block 3-connected component

RESULT: the size of the 2-connected core

2 different behaviours for graphs WITH core

  • Subcritical scheme (Series-parallel like): discrete law.

q(u) =

∞

  • k=0

p(Xn = k)uk = τuB′′(uτ), τB′′(τ) = 1.

  • Scheme 3

2 ◦ 3 2 (Planar like): two regimes

  • for fixed k, p(Xn = k) follows a discrete limit law.
  • For k = αn + xn2/3, x = O(1), p(Xn = k) = n−2/3cg(cx).

We deduce also the size of the LARGEST block.

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Background Enumeration Largest block 3-connected component

RESULT: the size of the 2-connected core

2 different behaviours for graphs WITH core

  • Subcritical scheme (Series-parallel like): discrete law.

q(u) =

∞

  • k=0

p(Xn = k)uk = τuB′′(uτ), τB′′(τ) = 1.

  • Scheme 3

2 ◦ 3 2 (Planar like): two regimes

  • for fixed k, p(Xn = k) follows a discrete limit law.
  • For k = αn + xn2/3, x = O(1), p(Xn = k) = n−2/3cg(cx).

We deduce also the size of the LARGEST block.

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Background Enumeration Largest block 3-connected component

2 different pictures

Series-parallel-like situation Planar-like situation

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Background Enumeration Largest block 3-connected component

α in the planar case

Varying the density of edges we always get a critical composition scheme

1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0

The usual density is µ = 2,2132, with α = 0,9598.

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Background Enumeration Largest block 3-connected component

Largest 3-connected component

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Background Enumeration Largest block 3-connected component

The core in non-separable maps

Let C, M3 be the set of non-separable and 3-connected maps, respectively. C → C(z) = 2z2 + z2 + 2z3 + 6z4 + 22z5 + . . . M3 → M3(z) = z5 + 4z7 + 6z8 + 24z9 + . . . A composition scheme also holds in this context: C(z) = C(z) z − 2 ⇒ C(u, z) =

Graphs with core

  • M3(uC(z))

+

Coreless graphs

  • z + 2C(z)2

1 + C(z) The size of the core obeys an Airy distribution

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Background Enumeration Largest block 3-connected component

The core in non-separable maps

Let C, M3 be the set of non-separable and 3-connected maps, respectively. C → C(z) = 2z2 + z2 + 2z3 + 6z4 + 22z5 + . . . M3 → M3(z) = z5 + 4z7 + 6z8 + 24z9 + . . . A composition scheme also holds in this context: C(z) = C(z) z − 2 ⇒ C(u, z) =

Graphs with core

  • M3(uC(z))

+

Coreless graphs

  • z + 2C(z)2

1 + C(z) The size of the core obeys an Airy distribution

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SLIDE 61

Background Enumeration Largest block 3-connected component

The 3-connected core of a general map

Taking all the equations together: M(z) = C

  • z(1 + M(z))2

C(z) = C(z)/z − 2 C(z) = M3(C(z)) + z + 2C(z)2

1+C(z)

     ⇓ M(v, z) = z(1+M(z))2M3

  • vC(z(1 + M(z))2)
  • +Coreless graphs

Singular expansion: ⇒ 3

2 ◦ 3 2 composition scheme⇒ AIRY Law

QUESTION: Can we say something similar for graphs?

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SLIDE 62

Background Enumeration Largest block 3-connected component

Largest 3-connected core

The 3-connected components of a 2-connected graph are those 3-connected graphs that are the support of h-networks in the network decomposition Tz(x0, uD(x0, y)), for a good choice of x0. We can extend the definition to connected graphs

  • No explicit composition scheme: C(u, z) = A (uB(C(z))).
  • Parameters in terms of edges and vertices.
  • 2 variables: vertices and edges.
  • Which x0 is the good one?
slide-63
SLIDE 63

Background Enumeration Largest block 3-connected component

Largest 3-connected core

The 3-connected components of a 2-connected graph are those 3-connected graphs that are the support of h-networks in the network decomposition Tz(x0, uD(x0, y)), for a good choice of x0. We can extend the definition to connected graphs

  • No explicit composition scheme: C(u, z) = A (uB(C(z))).
  • Parameters in terms of edges and vertices.
  • 2 variables: vertices and edges.
  • Which x0 is the good one?
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SLIDE 64

Background Enumeration Largest block 3-connected component

RESULT: size of the largest 3-connected component

Let Xn be the number of vertices in the largest 3-connected component of a random connected planar graph with n vertices. Then p

  • Xn = α2n + xn2/3

∼ n−2/3c2g(c2x), where α2 ≈ 0,7346 and c2 ≈ 3,14596 are computable constants. A similar result (with different constants) is true for all planar-like families.

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SLIDE 65

Background Enumeration Largest block 3-connected component

The strategy of the proof

3 steps to get the result

N A A

conn. 2 − conn. 2 − conn. 2 − conn.

N

3 − conn. 3 − conn. vertices vertices edges vertices edges edges

AIRYdistributed

Asympt.equal AIRYdistributed AIRYdistributed

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SLIDE 66

Background Enumeration Largest block 3-connected component

The strategy of the proof

3 steps to get the result

N A A

AIRYdistributed conn. 2 − conn. 2 − conn. 2 − conn.

N

3 − conn. 3 − conn. vertices AIRYdistributed vertices edges vertices edges edges

AIRYdistributed

Asympt.equal

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SLIDE 67

Background Enumeration Largest block 3-connected component

The strategy of the proof

3 steps to get the result

N A A

AIRYdistributed Asympt.equal conn. 2 − conn. 2 − conn. 2 − conn.

N

3 − conn. 3 − conn. vertices AIRYdistributed vertices edges vertices edges edges

AIRYdistributed

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SLIDE 68

Background Enumeration Largest block 3-connected component

The strategy of the proof

3 steps to get the result

N A A

AIRYdistributed Asympt.equal conn. 2 − conn. 2 − conn. 2 − conn.

N

3 − conn. 3 − conn. vertices

AIRYdistributed

AIRYdistributed vertices edges vertices edges edges

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SLIDE 69

Background Enumeration Largest block 3-connected component

THANK YOU

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SLIDE 70

Background Enumeration Largest block 3-connected component

Graph classes with given 3-connected components: asymptotic enumeration and random graphs

Juanjo Ru´ e

(joint work with Omer Gim´ enez and Marc Noy)

Laboratoire d’Informatique, ´ Ecole Polytechnique,

Workshop on Random Graphs and Maps on Surfaces, IHP, 2nd November 2009