Graph classes with given 3-connected components:asymptotic counting and critical phenomena

Omer Gimenez (1), Marc Noy (2), Juanjo Ru e (2)

Universitat Politecnica de Catalunya (UPC)

(1) Departament de Llenguatges i Sistemes Informatics (LSI)

(2) Departament de Matematica Aplicada 2 (MA2)

Graph classes with given 3-connected components – p. 1

This talk

• Families of graphs with given 3-connected components◦ Relation with minor-free graphs

• Asymptotic enumeration dichotomy;◦ SP-like families of graphs◦ Planar-like families of graphs

• Critical phenomena on graphs with fixed density of edges• Scope: size of the 2, 3-connected core, . . .

Graph classes with given 3-connected components – p. 2

Objects: simple labelled graphs

• simple : no loops, no multiple edges.• labelled : vertices are labelled.

• graphs : no embedding.

Graph classes with given 3-connected components – p. 3

Tools (I): the symbolic method

Translate set operations in combinatorial structures into formaloperations between GF.

In our problem:

• an,m: # of elements with n vertices and m edges.

• Generating function A(x, y) :◦ The variable x counts the vertices, y counts the edges.◦ The GFs are exponential on x and ordinary on y.

A(x, y) =∑

n,m≥0

an,mxn

n!ym

• Univariate generating function A(x) = A(x, 1) =∑

n≥0an

xn

n!

Graph classes with given 3-connected components – p. 4

Tools (II): Singularity Analysis

The smallest singularity ρ of A(x) determines the asymptotics of an.

• Location of ρ −→ exponential growth constant γ.• Behaviour of A(x) at ρ −→ subexponential terms in asymptotics

of an.

Transfer Theorems (Flajolet, Odlyzko)

Let α /∈ {0,−1,−2, . . .}. If

A(x) ∼x→ρ a · (1− x/ρ)−α ⇐⇒ A(x) = a · (1− x/ρ)−α + o((1− x/ρ)−α)

then

gn ∼g

Γ(α)· nα−1 · ρ−n · n!

Graph classes with given 3-connected components – p. 5

General graphs from connected graphs

Let C be a family of connected graphs.

We define G as those graphs such that their connected componentsare in C.

G = SET(C) =⇒ G(x) = exp(C(x))

Graph classes with given 3-connected components – p. 6

General graphs from connected graphs

Let C be a family of connected graphs.

We define G as those graphs such that their connected componentsare in C.

G = SET(C) =⇒ G(x) = exp(C(x))

Graph classes with given 3-connected components – p. 6

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs.

We define C as those connected graphs such that their 2-connectedblocks are in B.

Graph classes with given 3-connected components – p. 7

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs.

We define C as those connected graphs such that their 2-connectedblocks are in B.

Graph classes with given 3-connected components – p. 7

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs.

We define C as those connected graphs such that their 2-connectedblocks are in B.

Graph classes with given 3-connected components – p. 7

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs.

We define C as those connected graphs such that their 2-connectedblocks are in B.

Graph classes with given 3-connected components – p. 7

Connected graphs from 2-connected graphs

Let B be a family of 2-connected graphs.

We define C as those connected graphs such that their 2-connectedblocks are in B.

In other words, a vertex-rooted connected graph is a tree of2-connected blocks.

Co = SET(Bo(v ← Co)) =⇒ xC′(x) = x expB′(xC′(x))

Graph classes with given 3-connected components – p. 7

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 obtainedfrom series, parallel, and T -compositions.

D(x, y) = (1 + y) exp

(

xD2

1 + xD+

1

2x2

∂T

∂z(x, D)

)

− 1

∂B

∂y(x, y) =

x2

2

(

1 + D(x, y)

1 + y

)

where D is the GF for networks (essentially edge-rooted 2-connectedgraphs without the edge root).

Graph classes with given 3-connected components – p. 8

Summing up: 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′(xC′(x, y), y))

G(x, y) = exp(C(x, y))

Graph classes with given 3-connected components – p. 9

Examples of families & excluded minors (I)

Examples.

• Series-parallel graphs

◦ Excluded minors:◦ Allowed 3-connected components: None.◦ T (x, z) = 0.

• Planar graphs

◦ Excluded minors: , ,◦ Allowed 3-connected components: 3-connected planar

graphs.◦ T (x, z): The number of labelled 2-connected planar graphs

[Bender, Gao, Wormald; 02]

Graph classes with given 3-connected components – p. 10

Examples of families & excluded minors (II)

• W4-free

◦ Excluded minors:

◦ Allowed 3-connected components:◦ T (x, z) = 1

4!x4z6.

• K−5

-free

◦ Excluded minors:

◦ Allowed 3-connected components: , ,

, , . . .◦ T (x, z) = 70

6!x6z9 − 1

2x

(

log(1− xz2) + 2xz2 + x2z4)

.

Graph classes with given 3-connected components – p. 11

Examples of families & excluded minors (III)

• K3,3-free [Gerke, G, Noy, Weibl; 06]

◦ Excluded minors:

◦ Allowed 3-connected components: , 3-connected planarmaps.

◦ T (x, z) = . . . .

• If G = Ex(M) and all the excluded minorsM are 3-connected,then G can be expressed in terms of its 3-connected graphs.

• Problem: finding the set of allowed 3-connected components.

Graph classes with given 3-connected components – p. 12

Example: G =Planar graphs

• Embeddable in the sphere.• Do not contain K5, K3,3 as a minor.

Asymptotic enumeration and limit laws of planar graphs [G., Noy; 05]

Graph classes with given 3-connected components – p. 13

Example: G =Planar graphs

• Embeddable in the sphere.• Do not contain K5, K3,3 as a minor.

Asymptotic enumeration:

bn ∼ b · n−7/2 · γnB · n!

cn ∼ c · n−7/2 · γn · n!

gn ∼ g · n−7/2 · γn · n!

γB ≃ 26.1841

γ ≃ 27.2268

Uniformly distributed random graph Gn:

Edges(Gn) ∼ N(λn, σ2n)

Components(Gn) ∼ 1 + P (ν)

λ ≃ 2.2132

ν ≃ 0.0374

Graph classes with given 3-connected components – p. 13

Example: G =Series-parallel graphs

• Tree-width smaller or equal than 2.• Obtained from acyclic graph by series and parallel operations.

On the number of series-parallel and outerplanar graphs [Bodirsky, G,Kang, Noy]

Graph classes with given 3-connected components – p. 14

Example: G =Series-parallel graphs

• Tree-width smaller or equal than 2.• Obtained from acyclic graph by series and parallel operations.

Asymptotic enumeration:

bn ∼ b · n−5/2 · γnB · n!

cn ∼ c · n−5/2 · γn · n!

gn ∼ g · n−5/2 · γn · n!

γB ≃ 7.8125

γ ≃ 9.0735

Uniformly distributed random graph Gn:

Edges(Gn) ∼ N(λn, σ2n)

Components(Gn) ∼ 1 + P (ν)

λ ≃ 1.6167

ν ≃ 0.1176

Graph classes with given 3-connected components – p. 14

Equations

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′(xC′(x, y), y))

G(x, y) = exp(C(x, y))

Planar graphs

u(x, z) = xz(1 + v(x, z))2

v(x, z) = z(1 + u(x, z))2

∂T

∂z(x, z) = x2z2

(

1

1 + xz+

1

1 + z− 1− (1 + u)2(1 + v)2

(1 + u + v)3

)

Graph classes with given 3-connected components – p. 15

Equations

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′(xC′(x, y), y))

G(x, y) = exp(C(x, y))

Series-parallel graphs{

∂T

∂z(x, z) = 0

Graph classes with given 3-connected components – p. 15

Equations

Series-parallel graphs{

∂T

∂z(x, z) = 0

Planar graphs

u(x, z) = xz(1 + v(x, z))2

v(x, z) = z(1 + u(x, z))2

∂T

∂z(x, z) = x2z2

(

1

1 + xz+

1

1 + 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′(xC′(x, y), y))

G(x, y) = exp(C(x, y))

Graph classes with given 3-connected components – p. 16

Equations

Series-parallel graphs{

∂T

∂z(x, z) = 0

Planar graphs

u(x, z) = xz(1 + v(x, z))2

v(x, z) = z(1 + u(x, z))2

∂T

∂z(x, z) = x2z2

(

1

1 + xz+

1

1 + z+ . . .

∂T

∂z∼ 0

∂T

∂z∼ t · (1− z

z0

)3/2

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′(xC′(x, y), y))

G(x, y) = exp(C(x, y))

Graph classes with given 3-connected components – p. 16

Equations

Series-parallel graphs{

∂T

∂z(x, z) = 0

Planar graphs

u(x, z) = xz(1 + v(x, z))2

v(x, z) = z(1 + u(x, z))2

∂T

∂z(x, z) = x2z2

(

1

1 + xz+

1

1 + z+ . . .

∂T

∂z∼ 0

∂T

∂z∼ t · (1− z

z0

)3/2

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′(xC′(x, y), y))

G(x, y) = exp(C(x, y))

D ∼ d · (1− x

x0

)1/2

B ∼ b · (1− x

x0

)3/2

C ∼ c · (1− x

ρ)3/2

G ∼ g · (1− x

ρ)3/2

D ∼ d · (1− x

x0

)3/2

B ∼ b · (1− x

x0

)5/2

C ∼ c · (1− x

ρ)5/2

G ∼ g · (1− x

ρ)5/2

Graph classes with given 3-connected components – p. 16

Results

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−n0· n!

bn ∼ b · n−5/2 · x−n0· n!

cn ∼ c · n−5/2 · ρ−n · n!

gn ∼ g · n−5/2 · ρ−n · n!

Graph classes with given 3-connected components – p. 17

Results

If ∂T∂z (x, z) has singularity type (1− z/z0)

3/2, then 3 different situationsmay 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−n0· n!

bn ∼ b · n−7/2 · x−n0· n!

cn ∼ c · n−7/2 · ρ−n · n!

gn ∼ g · n−7/2 · ρ−n · n!

Graph classes with given 3-connected components – p. 18

Results

If ∂T∂z (x, z) has singularity type (1− z/z0)

3/2, then 3 different situationsmay 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−n0· n!

bn ∼ b · n−5/2 · x−n0· n!

cn ∼ c · n−5/2 · ρ−n · n!

gn ∼ g · n−5/2 · ρ−n · n!

Graph classes with given 3-connected components – p. 18

Results

If ∂T∂z (x, z) has singularity type (1− z/z0)

3/2, then 3 different situationsmay 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−n0· n!

bn ∼ b · n−7/2 · x−n0· n!

cn ∼ c · n−5/2 · ρ−n · n!

gn ∼ g · n−5/2 · ρ−n · n!

Graph classes with given 3-connected components – p. 18

Asymptotic enumeration

Some parameters can be expressed in terms of T (x, z):

• Growth constants x0 (2-connected) and ρ (connected, all graphs).• Asymptotic constants d, b, c, g.• λ and σ of the Normally distributed edges.• Parameter ν of the Poisson distributed connected components.

Graph classes with given 3-connected components – p. 19

Graphs with fixed density of edges

• Almost all graphs in G have the same edge density λ.• What about graphs in G with other edge densities?◦ Singularity analysis of G(x, y) when y 6= 1

Asymptotic enumeration of graphs with (λn−√n, λn +√

n) edges:

g(λ) · nα · ρ(λ)−n · n!

where α = −5/2 (series-parallel-like case) or α = −7/2 (planar -likecase).

Graph classes with given 3-connected components – p. 20

Critical phenomena

The singularity type of G(x, y) may change with the value of y.

Example: T = {3-connected cubic planar graphs}.

• c(λ)n: connected graphs of G with (λn−√n, λn +√

n) edges.

◦ If λ < 1.1844 . . .,

c(λ)n ∼ cλ · n−7/2 · ρ(λ)−n · n!

◦ If λ > 1.1844 . . .,

c(λ)n ∼ cλ · n−5/2 · ρ(λ)−n · n!

Graph classes with given 3-connected components – p. 21

Critical phenomena

The singularity type of G(x, y) may change with the value of y.

Example: T = {3-connected cubic planar graphs}.

0

0.2

0.4

0.6

0.8

1

1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18

• (top) Probability of small core• (bottom) Size of 2-connected large core

Graph classes with given 3-connected components – p. 21

Critical phenomena

The singularity type of G(x, y) may change with the value of y.

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 . . .,

b(λ)n ∼ bλ · n−7/2 · x0(λ)−n · n!

◦ If λ > 1.31725 . . .,

b(λ)n ∼ bλ · n−5/2 · x0(λ)−n · n!

Graph classes with given 3-connected components – p. 21

Critical phenomena

The singularity type of G(x, y) may change with the value of y.

Example: T = {3-connected cubic planar graphs}.

0

0.2

0.4

0.6

0.8

1

1.05 1.1 1.15 1.2 1.25 1.3

• (top) Probability of small core• (bottom) Number of edges of 3-connected large core

Graph classes with given 3-connected components – p. 21