Asymptotic study of subcritical graph classes Michael Drmota, Eric - - PowerPoint PPT Presentation

Asymptotic study of subcritical graph classes

Michael Drmota, Eric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Ru´ e

Laboratoire d’Informatique, ´ Ecole Polytechnique, ERC Exploremaps Project

VII Jornadas de Matem´ atica Discreta y Algor´ ıtmica, Castro Urdiales, 7 Julio 2010

The material of this talk

1.− Background and notation. 2.− Naive description of the graphs we want to enumerate: subcritical graph families. 3.− The strategy: graph decompositions, the grammar and functional system of equations. 4.− Results, and explicit computations. 5.− Further research and open problems.

Background

Objects: graphs

Labelled Graph= labelled vertices+edges. Unlabelled Graph= labelled one up to permutation of labels. Simple Graph= NO multiples edges, NO loops.

3 1 2 3 1 2 1 2 3 3 2 1

Question: How many graphs with n vertices are in the family?

The counting series

Strategy: Encapsulate these numbers → Counting series

◮ Labelled framework: exponential generating functions

A(x) = ∑

a∈A

x|a| |a|! =

∞

∑

n≥0

|An| n! xn

◮ Unlabelled framework: cycle index sums

ZA(s1, s2, . . .) = ∑

n≥0

1 n! ∑

(σ,g)∈Sn×An σ·g=g

sc1

1 sc2 2 · · · scn n ,

• A(x) = ZA(x, x2, x3, . . .) =

∑

n≥0

| An|xn.

The symbolic method

COMBINATORIAL RELATIONS between CLASSES ↕⇕↕ EQUATIONS between GENERATING FUNCTIONS Class Labelled setting Unlabelled setting C = A ∪ B C(x) = A(x) + B(x)

• C(x) =

A(x) + B(x) C = A × B C(x) = A(x) · B(x)

• C(x) =

A(x) · B(x) C = Set(B) C(z) = exp(B(x))

• C(x) = exp

( ∑

i≥1 1 i

B(xi) ) C = A ◦ B C(x) = A(B(x))

• C(x) = ZA(

B(x), B(x2), . . .)

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: subexponential growth ◮ Transfer Theorems: Let α /

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

Limit laws

Study of parameters→ A(u, z) = ∑∞

n,m=0 an,mznum.

For a fixed n, the numbers an,m describe 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

Families of graphs under study

The main construction

We use easier graphs as fundamental pieces. 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. C• = v × Set(B′(v ← C•))

Some families of graphs (1)

• 1. Plane trees (Ex(K3)):
• 1. Explicit expressions

Some families of graphs (2)

• 2. Cacti graphs:
• 1. Explicit expressions

ZB′ (s1, s2, . . .) = s1 + s2

1

2(1 − s1) + 1 + s1 2(1 − s2) ,

Some families of graphs (3)

• 3. Outerplanar graphs (Ex(K4, K2,3)):
• 1. Explicit expressions (Bodirsky, Fusy, Kang, Vigerske, 2007)

ZB(s1, . . . ) = − 1 2 ∑

d>0

ϕ(d) d log ( 3 4 − 1 4 sd + 1 4 √ s2

d − 6sd + 1

) + s2 + s2

1 − 4s1 − 2

16 + s2

1 − 3s2 1s2 + 2s1s2

16s2

2

+ 3 − s1 16 √ s2

1 − 6s1 + 1 −

1 16 ( 1 + s2

1

s2

2

+ 2 s1 s2 ) √ s2

2 − 6s2 + 1

Some families of graphs (and 4)

• 4. Series-parallel graphs (Ex(K4)):
• 5. NO explicit expressions !

The subcritical condition

All the previous families are defined in the following way: C• = v × Set(B′(v ← C•)) Which translates into the equations C•(x) = x exp(B′(C•(x))),

• C•(x) = x exp

(∑

i≥1 1 i ZB′(

C•(xi), C•(x2i), . . . ) ) . In both cases, the counting series for connected graphs is determined by the counting series for 2-connected ⇓ Subcritical condition The singularity for the connected counting series is related to a branch point (derivative equals to 0) of the 2-connected counting series.

Graph decomposition, a grammar and system of functional equations

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))

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))

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. Co = v × Set(Bo(v ← Co)) = ⇒ xC′(x) = x exp B′(xC′(x))

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. C• = v × SET(Bo(v ← C•)) = ⇒ xC′(x) = x exp B′(xC′(x))

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)

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

A set of 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′(C•(x, y), y) ) G(x, y) = exp(C(x, y))

But in the unlabelled framework, things are more involved...

The complete grammar

A Grammar for Decomposing a Family of Graphs into 3-connected Components; Chapuy, Fusy, Kang, Shoilekova

This system is obtained ap- plying the dissymmetry theorem for trees in an ingenious way. Hence, in the unlabelled framework we need to study system of functional equations more involved.

A system of functional equations

If a combinatorial system of equations is “regular” enough, we can assure square-root developements [Drmota, 1997] Consider the functional system of equations y = F(y; z, v). If the system satisfies some “‘nice” conditions at v = v0, then, around v = v0

◮ There is a unique vector of power series y = y(z, v) in the

variables z, v that satisfies the system.

◮ The components of y have non-negative coefficients

[zn] yi(z, v0) (for i ∈ {1, . . . , r}).

◮ The components of y have a square-root expansion

around (z0, v0). ⇓ Expansions of the form c · n−3/2ρ−n(1 + o(1)).

Results and explicit values

Asymptotic enumeration

Our main result is the following one: [Drmota, Fusy, Kang, Kraus R., 2010] Let G be a subcritical block-stable graph class (either labelled

• r unlabelled). Then,

[zn]C•(z) = c1 n−3/2 γn(1 + o(1)), [zn]G(z) = c2 n−5/2 γn(1 + o(1)), and for certain constants c1, c2, γ. Exponent n−3/2 = arborescent structure= branch point.

Limit laws (1)

We study parameters on a random connected graph with n vertices: number of edges, number of cut-vertices, number of blocks. In all cases (independently of the framework) we get Xn − E Xn √Var Xn → N(0, 1), Problem: it is usual that we do not know to prove that Var Xn ̸= 0 without explicit computation ⇓ We find a general analytic criteria on the counting series which assures that Var Xn ̸= 0.

Limit laws (2)

We study the Degree distribution

◮ Xk n: number of vertices of degree k in a randomly chosen

graph with n vertices.

◮ dk the limiting probability that the root vertex of a

randomly chosen graph is k. We show the following:

◮ We get closed expressions for dk. ◮ Xk n has a normal limiting distribution.

A numerical table

Constant growth for different subcritical graph families Family Labelled Unlabelled Acyclic 2,71828 2,95577 Cacti 4,18865 4,50144 Outerplanar 7,32708 7,50360 Series-Parallel 9,07359 9.38527 The constant growth for unlabelled SP-graphs has been

• btained using

◮ Generation of the first terms of the counting series. ◮ Approximating the system of equations by another easier. ◮ Checking convergence of the singular point associated to

the system (Pivoteau, Salvy, Soria)

Open problems

Beyond the subcritical scheme

The subcritical condition can be solved easily, compared with a critical condition. Families of graphs which arise from the map context do not satisfy a subcritical condition ⇓ Next step: take a natural family of 3-connected graphs arising from maps (triangulations), and study the critical scheme.

The enumeration of unlabelled planar graphs

The problem was completely solved by Gim´ enez and Noy, but little is known in the unlabelled setting:

◮ It is necessary to obtain the counting series for 3 connected

planar maps (unlabelled): exact enumeration of 3-polytopes.

◮ It is necessary to deal with these counting series! ◮ It is necessary to solve the critical problem to get the

enumeration of vertex-rooted connected planar graphs. Lot of work to be done!

Thank you

Asymptotic study of subcritical graph classes

Michael Drmota, Eric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Ru´ e

Laboratoire d’Informatique, ´ Ecole Polytechnique, ERC Exploremaps Project

VII Jornadas de Matem´ atica Discreta y Algor´ ıtmica, Castro Urdiales, 7 Julio 2010