victor zamaraev – boundary properties of factorial classes of graphs

Boundary properties of factorial classes of graphs

Victor Zamaraev

Laboratory of Algorithms and Technologies for Networks Analysis (LATNA),Higher School of Economics

Joint work withVadim Lozin, University of Warwick

Workshop on Extremal Graph Theory6 June 2014

Introduction

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Introduction

All considered graphs are simple (undirected, without loops andwithout multiple edges).

Graphs are labeled by natural numbers 1, . . . , n

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Introduction

All considered graphs are simple (undirected, without loops andwithout multiple edges).

Graphs are labeled by natural numbers 1, . . . , n

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Introduction

Definition

A class is a set of graphs closed under isomorphism.

Definition

A class of graphs is hereditary if it is closed under taking inducedsubgraphs.

ExapmleLet X be a hereditary class and W4 ∈ X. Then C4 ∈ X.

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Introduction

Definition

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Introduction

Definition

W4 C4 4 / 28

Introduction

Every hereditary graph class X can be defined by a set offorbidden induced subgraphs.

LetM be a set of graphs. Then Free(M) denotes the set of allgraphs not containing induced subgraphs isomorphic to graphs fromM.

Statement

Class X is hereditary if and only if there existsM such thatX = Free(M).

We say that graphs in X areM-free.

Example

For the class of bipartite graphsM is {C3, C5, C7, . . . }, i.e.B = Free(C3, C5, C7, . . . ).

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Introduction

Statement

Example

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Introduction

Statement

Example

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Introduction

Statement

Example

For the class of bipartite graphsM is {C3, C5, C7, . . . }, i.e.B = Free(C3, C5, C7, . . . ). 5 / 28

Introduction

For a class X denote by Xn the set of n-vertex graphs from X.

Example

Let P be the class of all graph.

|Pn| = 2(n2) = 2n(n−1)/2

log2 |Pn| = Θ(n2)

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Introduction

Example

|Pn| = 2(n2) = 2n(n−1)/2

log2 |Pn| = Θ(n2)

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Introduction

Example

|Pn| = 2(n2) = 2n(n−1)/2

log2 |Pn| = Θ(n2)

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Introduction

Theorem (Alekseev V. E., 1992; Bollobas B. and Thomason A., 1994)

For every infinite hereditary class X, which is not the class of allgraphs:

log2 |Xn| =(

1− 1

2+ o(n2), (1)

where c(X) ∈ N is the index of class X.

(i) For c(X) > 1, log2 |Xn| = Θ(n2)

(ii) For c(X) = 1, log2 |Xn| = o(n2)

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Introduction

log2 |Xn| =(

1− 1

2+ o(n2), (1)

(i) For c(X) > 1, log2 |Xn| = Θ(n2)

(ii) For c(X) = 1, log2 |Xn| = o(n2)

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Introduction

log2 |Xn| =(

1− 1

2+ o(n2), (1)

(i) For c(X) > 1, log2 |Xn| = Θ(n2)

(ii) For c(X) = 1, log2 |Xn| = o(n2)

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Introduction

Let c(X) = 1

Question

What are possible rates of growth of a function log2 |Xn|?

Scheinerman E.R., Zito J. (1994)

Constant classes: log2 |Xn| = Θ(1).Polynomial classes: log2 |Xn| = Θ(log n).Exponential classes: log2 |Xn| = Θ(n).Factorial classes: log2 |Xn| = Θ(n log n).All other classes are superfactorial.

There are no intermediate growth rates between first four ranges.For exmaple, there is no hereditary class X withlog2 |Xn| = Θ(

√n).

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Introduction

Let c(X) = 1

Question

Constant classes: log2 |Xn| = Θ(1).

Polynomial classes: log2 |Xn| = Θ(log n).Exponential classes: log2 |Xn| = Θ(n).Factorial classes: log2 |Xn| = Θ(n log n).All other classes are superfactorial.

√n).

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Introduction

Let c(X) = 1

Question

Constant classes: log2 |Xn| = Θ(1).Polynomial classes: log2 |Xn| = Θ(log n).

Exponential classes: log2 |Xn| = Θ(n).Factorial classes: log2 |Xn| = Θ(n log n).All other classes are superfactorial.

√n).

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Introduction

Let c(X) = 1

Question

Constant classes: log2 |Xn| = Θ(1).Polynomial classes: log2 |Xn| = Θ(log n).Exponential classes: log2 |Xn| = Θ(n).

Factorial classes: log2 |Xn| = Θ(n log n).All other classes are superfactorial.

√n).

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Introduction

Let c(X) = 1

Question

Constant classes: log2 |Xn| = Θ(1).Polynomial classes: log2 |Xn| = Θ(log n).Exponential classes: log2 |Xn| = Θ(n).Factorial classes: log2 |Xn| = Θ(n log n).

All other classes are superfactorial.

√n).

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Introduction

Let c(X) = 1

Question

√n).

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Introduction

Let c(X) = 1

Question

√n).

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Introduction

Constant

Polynomial

Exponential

Factorial layer

Classes with index 1

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Introduction

Example

Constant class: Co – complete graphs (1).

Polynomial class: E1 – graphs with at most one edge((n2

)+ 1).

Exponential class: Co + Co (2n−1).Factorial class: F – forests (nn−2 < |Fn| < n2n).

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Introduction

Example

Constant class: Co – complete graphs (1).Polynomial class: E1 – graphs with at most one edge((n2

)+ 1).

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Introduction

Example

)+ 1).

Exponential class: Co + Co (2n−1).

Factorial class: F – forests (nn−2 < |Fn| < n2n).

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Introduction

Example

)+ 1).

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Introduction

Alekseev V.E. (1997)

1 Structural characterizations were obtained for the first threelayers.

2 In every of the four layers all minimal classes were found.

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Introduction

Alekseev V.E. (1997)

1 Structural characterizations were obtained for the first threelayers.

2 In every of the four layers all minimal classes were found.

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Introduction

Constant

Polynomial

Exponential

Factorial layer

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Introduction

Balogh J., Bollobas B., Weinreich D. (2000)

In addition1 Characterized lower part of the factorial layer, i.e. classes with|Xn| < n(1+o(1))n.

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Introduction

Balogh J., Bollobas B., Weinreich D. (2000)

In addition1 Characterized lower part of the factorial layer, i.e. classes with|Xn| < n(1+o(1))n.

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Introduction

Examples of factorial classes:

forestsplanar graphsline graphscographspermutation graphsthreshold graphsgraphs of bounded vertex degreegraphs of bounded clique-widthet al.

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Introduction

Problem

Characterize factorial layer.

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Minimal superfactorial classes

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Constant

Polynomial

Exponential

Factorial

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Constant

Polynomial

Exponential

Factorial

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log2 |Xn| = Θ(n log2 n)

CB = Free(C3, C5, C6, C7, . . .)

Theorem (Spinrad J. P., 1995)

log2 |CBn| = Θ(n log2 n)

Question

Is the class of chordal bipartite graphs a minimal superfactorial?

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CB = Free(C3, C5, C6, C7, . . .)

Question

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CB = Free(C3, C5, C6, C7, . . .)

Question

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CB = Free(C3, C5, C6, C7, . . .)

Question

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Theorem (Dabrowski K., Lozin V.V., Zamaraev V., 2012)

Let X = Free(2C4, 2C4 + e)∩CB. Then log2 |Xn| = Θ(n log2 n).

2C4 2C4 + e

Open question

Is the class Free(2C4, 2C4 + e) ∩CB a minimal superfactorial?

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Theorem (Dabrowski K., Lozin V.V., Zamaraev V., 2012)

Let X = Free(2C4, 2C4 + e)∩CB. Then log2 |Xn| = Θ(n log2 n).

2C4 2C4 + e

Open question

Is the class Free(2C4, 2C4 + e) ∩CB a minimal superfactorial?

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Denote by Bk the class of (C4, C6, ..., C2k)-free bipartite graphs.

Statement (follows from the results of Lazebnik F., et al., 1995)

For each integer k ≥ 2, the class Bk is superfactorial.

Infinite sequence of superfactorial classes

B2 ⊃ B3 ⊃ B4 . . . .

In this sequence there is no minimal superfactorial class.

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B2 ⊃ B3 ⊃ B4 . . . .

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B2 ⊃ B3 ⊃ B4 . . . .

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B2 ⊃ B3 ⊃ B4 . . . .

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Limit classes

Definition

Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we willsay that the sequence converges to a class X if

⋂i≥1

Xi = X.

ExampleThe sequence B2 ⊃ B3 ⊃ B4 . . . converges to the factorial class Fof forests, i.e.

⋂i≥1

Bi = F.

Definition

A class X of graphs is a limit class (for the factorial layer) if thereis a sequence of superfactorial classes converging to X.

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Limit classes

Definition

⋂i≥1

Xi = X.

⋂i≥1

Bi = F.

Definition

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Limit classes

Definition

⋂i≥1

Xi = X.

⋂i≥1

Bi = F.

Definition

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Boundary classes

Definition

A limit class is called boundary (or minimal) if it does not properlycontain any other limit class.

Theorem

A finitely defined class is superfactorial if and only if it contains aboundary class.

Theorem

The class of forests is a boundary class.

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Boundary classes

Definition

Theorem

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Boundary classes

Definition

Theorem

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Are there more boundary classes?

There are five more boundary classes, which can be easly obtainedfrom the class of forests.

Two of them are:1 complements of forests;2 bipartite complements of forests;

Bipartite complement of F

Question

Are there other boundary classes?

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Question

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Question

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Question

Are there other boundary classes?23 / 28

Lozin’s conjecture

Conjecture (Lozin’s conjecture, [Lozin V.V., Mayhill C., Zamaraev V., 2011])

A hereditary graph class X is factorial if and only if at least one ofthe following three classes: X ∩B, X ∩ B и X ∩ S is factorial andeach of these classes is at most factorial.

B – bipartite graphsB – complements of bipartite graphsS – split graphs

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Proper boundary subclasses

B2 = Free(C4) ∩B CB = Free(C3, C5, C6, . . .)

superfactorial superfactorial⋂i≥1

Bi = F ⊂ B2⋂i≥1

Bi = F ⊂ CB

Bi ⊆ B2, i ≥ 1 Bi * CB, i ≥ 1

Definition

Let X be a superfactorial class and S a boundary subclasscontained in X. We say that S is a proper boundary subclass of Xif there is a sequence of superfactorial subclasses of X convergingto S.

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superfactorial superfactorial

⋂i≥1

Bi = F ⊂ B2⋂i≥1

Bi = F ⊂ CB

Bi ⊆ B2, i ≥ 1 Bi * CB, i ≥ 1

Definition

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Bi = F ⊂ B2⋂i≥1

Bi = F ⊂ CB

Bi ⊆ B2, i ≥ 1 Bi * CB, i ≥ 1

Definition

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Bi = F ⊂ B2⋂i≥1

Bi = F ⊂ CB

Bi ⊆ B2, i ≥ 1 Bi * CB, i ≥ 1

Definition

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Bi = F ⊂ B2⋂i≥1

Bi = F ⊂ CB

Bi ⊆ B2, i ≥ 1 Bi * CB, i ≥ 1

Definition

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Theorem

There are no proper boundary subclasses of chordal bipartitegraphs.

Theorem

The class of forests is the only proper boundary subclass of B2.

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Theorem

There are no proper boundary subclasses of chordal bipartitegraphs.

Theorem

The class of forests is the only proper boundary subclass of B2.

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Open problems

Open question

Find a minimal superfactorial class.

Open question

Is the list of boundary classes we found complete?

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Open problems

Open question

Find a minimal superfactorial class.

Open question

Is the list of boundary classes we found complete?

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Thank you!

victor zamaraev – boundary properties of factorial classes of graphs

Science

class of graphs

set of graphs

vertex graphs

class of bipartite graphs

hereditary graph class

constant classes

statement class x

polynomial classes