graph classes with given 3-connected components ... · on the number of series-parallel and...
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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