The Micro-World of Cographs

The Micro-world of Cographs

Bogdan Alecu Vadim Lozin Dominique de Werra

June 10, 2020

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 1 / 26 Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G ∈ X , and unbounded in X otherwise.

We work with ﬁnite, simple, undirected graphs. A hereditary class (just “class” from now on) is a set of graphs closed under taking induced subgraphs.

Basic deﬁnitions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G ∈ X , and unbounded in X otherwise.

A hereditary class (just “class” from now on) is a set of graphs closed under taking induced subgraphs.

Basic deﬁnitions

We work with ﬁnite, simple, undirected graphs.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G ∈ X , and unbounded in X otherwise.

Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S.

Basic deﬁnitions

We work with ﬁnite, simple, undirected graphs. A hereditary class (just “class” from now on) is a set of graphs closed under taking induced subgraphs.

Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S.

Basic deﬁnitions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G ∈ X , and unbounded in X otherwise.

A graph parameter is a function which associates to each graph a number.

Basic deﬁnitions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G ∈ X , and unbounded in X otherwise.

All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ...

Basic deﬁnitions

Examples: chromatic number, clique-width, ...

Basic deﬁnitions

We work with ﬁnite, simple, undirected graphs. A hereditary class (just “class” from now on) is a set of graphs closed under taking induced subgraphs. Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs.

Basic deﬁnitions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 Basic deﬁnitions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

An example: let ν(G) denote the number of vertices of G. Let K be the class of complete graphs, and K their complements. K and K are minimal classes of unbounded ν. In fact, they are the only minimal classes of unbounded ν. Theorem (Ramsey) A class X is of unbounded ν if and only if it contains K or K.

Some motivation

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 An example: let ν(G) denote the number of vertices of G. Let K be the class of complete graphs, and K their complements. K and K are minimal classes of unbounded ν. In fact, they are the only minimal classes of unbounded ν. Theorem (Ramsey) A class X is of unbounded ν if and only if it contains K or K.

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 Let K be the class of complete graphs, and K their complements. K and K are minimal classes of unbounded ν. In fact, they are the only minimal classes of unbounded ν. Theorem (Ramsey) A class X is of unbounded ν if and only if it contains K or K.

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

An example: let ν(G) denote the number of vertices of G.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 K and K are minimal classes of unbounded ν. In fact, they are the only minimal classes of unbounded ν. Theorem (Ramsey) A class X is of unbounded ν if and only if it contains K or K.

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

An example: let ν(G) denote the number of vertices of G. Let K be the class of complete graphs, and K their complements.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 In fact, they are the only minimal classes of unbounded ν. Theorem (Ramsey) A class X is of unbounded ν if and only if it contains K or K.

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

An example: let ν(G) denote the number of vertices of G. Let K be the class of complete graphs, and K their complements. K and K are minimal classes of unbounded ν.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 Theorem (Ramsey) A class X is of unbounded ν if and only if it contains K or K.

Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 Some motivation

An interesting problem: given a parameter p, can we characterise the classes in which p is bounded?

An idea: try to ﬁnd the “smallest” obstructions to boundedness. A class X is minimal of unbounded p if p is unbounded in X , but bounded in every proper subclass of X .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 3 / 26 let z(G) be the largest number n such that G has the cycle Cn as an induced subgraph, and let C be the hereditary closure of the set {C3, C4,... }. Then z is unbounded in C, but C has no minimal subclass of unbounded z. Proof: If z is unbounded in X ⊆ C, then X must contain inﬁnitely many cycles Ci . But then X ∩ Free(Ck ) (for any k with Ck ∈ X ) is a strictly smaller class in which z is unbounded.

A bypass: restrict ourselves to a setting in which this does not happen.

Some motivation

An obstacle:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 Then z is unbounded in C, but C has no minimal subclass of unbounded z. Proof: If z is unbounded in X ⊆ C, then X must contain inﬁnitely many cycles Ci . But then X ∩ Free(Ck ) (for any k with Ck ∈ X ) is a strictly smaller class in which z is unbounded.

A bypass: restrict ourselves to a setting in which this does not happen.

Some motivation

An obstacle: let z(G) be the largest number n such that G has the cycle Cn as an induced subgraph, and let C be the hereditary closure of the set {C3, C4,... }.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 Proof: If z is unbounded in X ⊆ C, then X must contain inﬁnitely many cycles Ci . But then X ∩ Free(Ck ) (for any k with Ck ∈ X ) is a strictly smaller class in which z is unbounded.

A bypass: restrict ourselves to a setting in which this does not happen.

Some motivation

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 If z is unbounded in X ⊆ C, then X must contain inﬁnitely many cycles Ci . But then X ∩ Free(Ck ) (for any k with Ck ∈ X ) is a strictly smaller class in which z is unbounded.

A bypass: restrict ourselves to a setting in which this does not happen.

Some motivation

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 But then X ∩ Free(Ck ) (for any k with Ck ∈ X ) is a strictly smaller class in which z is unbounded.

A bypass: restrict ourselves to a setting in which this does not happen.

Some motivation

An obstacle: let z(G) be the largest number n such that G has the cycle Cn as an induced subgraph, and let C be the hereditary closure of the set {C3, C4,... }. Then z is unbounded in C, but C has no minimal subclass of unbounded z. Proof: If z is unbounded in X ⊆ C, then X must contain inﬁnitely many cycles Ci .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 A bypass: restrict ourselves to a setting in which this does not happen.

Some motivation

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 Some motivation

A bypass: restrict ourselves to a setting in which this does not happen.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 4 / 26 There are no inﬁnite strictly descending chains (“well-foundedness”). There are no inﬁnite antichains.

The class of all graphs is not wqo by the induced subgraph relation. An inﬁnite antichain is given by the cycles C3, C4,... . The set of all classes of graphs is not wqo (not even well-founded) under inclusion. An inﬁnite strictly descending chain is given by Xi := Free(C3,..., Ci ), i ≥ 3.

Examples

Well-quasi-orderability

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 There are no inﬁnite strictly descending chains (“well-foundedness”). There are no inﬁnite antichains.

A chain is a set of elements of X , every two of which are ≤–comparable. An antichain is a set of elements of X , no two of which are ≤–comparable. (X , ≤) is well-quasi-ordered (“wqo” for short) if:

Examples

Well-quasi-orderability

Let ≤ be a quasi-order on a set X .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 There are no inﬁnite strictly descending chains (“well-foundedness”). There are no inﬁnite antichains.

An antichain is a set of elements of X , no two of which are ≤–comparable. (X , ≤) is well-quasi-ordered (“wqo” for short) if:

Examples

Well-quasi-orderability

Let ≤ be a quasi-order on a set X . A chain is a set of elements of X , every two of which are ≤–comparable.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 There are no inﬁnite strictly descending chains (“well-foundedness”). There are no inﬁnite antichains.

(X , ≤) is well-quasi-ordered (“wqo” for short) if:

Examples

Well-quasi-orderability

Let ≤ be a quasi-order on a set X . A chain is a set of elements of X , every two of which are ≤–comparable. An antichain is a set of elements of X , no two of which are ≤–comparable.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 The class of all graphs is not wqo by the induced subgraph relation. An inﬁnite antichain is given by the cycles C3, C4,... . The set of all classes of graphs is not wqo (not even well-founded) under inclusion. An inﬁnite strictly descending chain is given by Xi := Free(C3,..., Ci ), i ≥ 3.

There are no inﬁnite strictly descending chains (“well-foundedness”). There are no inﬁnite antichains.

Examples

Well-quasi-orderability

There are no inﬁnite antichains.

Examples

Well-quasi-orderability

Let ≤ be a quasi-order on a set X . A chain is a set of elements of X , every two of which are ≤–comparable. An antichain is a set of elements of X , no two of which are ≤–comparable. (X , ≤) is well-quasi-ordered (“wqo” for short) if: There are no inﬁnite strictly descending chains (“well-foundedness”).

Examples

Well-quasi-orderability

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 The set of all classes of graphs is not wqo (not even well-founded) under inclusion. An inﬁnite strictly descending chain is given by Xi := Free(C3,..., Ci ), i ≥ 3.

An inﬁnite antichain is given by the cycles C3, C4,... .

Well-quasi-orderability

Examples The class of all graphs is not wqo by the induced subgraph relation.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 An inﬁnite strictly descending chain is given by Xi := Free(C3,..., Ci ), i ≥ 3.

The set of all classes of graphs is not wqo (not even well-founded) under inclusion.

Well-quasi-orderability

Examples The class of all graphs is not wqo by the induced subgraph relation. An inﬁnite antichain is given by the cycles C3, C4,... .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 An inﬁnite strictly descending chain is given by Xi := Free(C3,..., Ci ), i ≥ 3.

Well-quasi-orderability

Examples The class of all graphs is not wqo by the induced subgraph relation. An inﬁnite antichain is given by the cycles C3, C4,... . The set of all classes of graphs is not wqo (not even well-founded) under inclusion.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 Well-quasi-orderability

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 5 / 26 Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

Otherwise, there exists Y1 ( Y which is of unbounded p. If Y1 is minimal, we are done. Otherwise, there exists Y2 ( Y1 which is of unbounded p. If Y2 is minimal, we are done. . . Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 Proof: If Y itself is minimal, we are done.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 If Y itself is minimal, we are done.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 Otherwise, there exists Y1 ( Y which is of unbounded p. If Y1 is minimal, we are done. Otherwise, there exists Y2 ( Y1 which is of unbounded p. If Y2 is minimal, we are done. . . Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 If Y1 is minimal, we are done. Otherwise, there exists Y2 ( Y1 which is of unbounded p. If Y2 is minimal, we are done. . . Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

Otherwise, there exists Y1 ( Y which is of unbounded p.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 Otherwise, there exists Y2 ( Y1 which is of unbounded p. If Y2 is minimal, we are done. . . Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

Otherwise, there exists Y1 ( Y which is of unbounded p. If Y1 is minimal, we are done.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 If Y2 is minimal, we are done. . . Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

Otherwise, there exists Y1 ( Y which is of unbounded p. If Y1 is minimal, we are done. Otherwise, there exists Y2 ( Y1 which is of unbounded p.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 . . Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

Otherwise, there exists Y1 ( Y which is of unbounded p. If Y1 is minimal, we are done. Otherwise, there exists Y2 ( Y1 which is of unbounded p. If Y2 is minimal, we are done.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 Since there are no inﬁnite strictly descending chains, this must terminate.

Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

Otherwise, there exists Y1 ( Y which is of unbounded p. If Y1 is minimal, we are done. Otherwise, there exists Y2 ( Y1 which is of unbounded p. If Y2 is minimal, we are done. . .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 Well-quasi-orderability

Theorem (folklore) A class X is wqo under the induced subgraph relation if and only if the set of subclasses of X is well-founded under inclusion.

Corollary Let X be a wqo class, Y ⊆ X a subclass, and let p be a parameter. If p is unbounded in Y, then there exists a minimal class Z ⊆ Y of unbounded p.

Proof: If Y itself is minimal, we are done.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 6 / 26 p is unbounded in Y.

There exists Z ∈ MX (p) such that Z ⊆ Y.

Corollary Let X be a wqo class, and let p be a parameter unbounded in X . Write MX (p) for the set of minimal subclasses of X where p is unbounded. Then for any subclass Y ⊆ X , the following are equivalent:

In other words, if X is wqo, then unboundedness in X of any parameter can be characterised in terms of minimal classes.

Well-quasi-orderability

This can be restated as follows:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 7 / 26 p is unbounded in Y.

There exists Z ∈ MX (p) such that Z ⊆ Y.

Then for any subclass Y ⊆ X , the following are equivalent:

In other words, if X is wqo, then unboundedness in X of any parameter can be characterised in terms of minimal classes.

Well-quasi-orderability

This can be restated as follows: Corollary Let X be a wqo class, and let p be a parameter unbounded in X . Write MX (p) for the set of minimal subclasses of X where p is unbounded.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 7 / 26 p is unbounded in Y.

There exists Z ∈ MX (p) such that Z ⊆ Y.

In other words, if X is wqo, then unboundedness in X of any parameter can be characterised in terms of minimal classes.

Well-quasi-orderability

This can be restated as follows: Corollary Let X be a wqo class, and let p be a parameter unbounded in X . Write MX (p) for the set of minimal subclasses of X where p is unbounded. Then for any subclass Y ⊆ X , the following are equivalent:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 7 / 26 There exists Z ∈ MX (p) such that Z ⊆ Y.

In other words, if X is wqo, then unboundedness in X of any parameter can be characterised in terms of minimal classes.

Well-quasi-orderability

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 7 / 26 In other words, if X is wqo, then unboundedness in X of any parameter can be characterised in terms of minimal classes.

Well-quasi-orderability

There exists Z ∈ MX (p) such that Z ⊆ Y.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 7 / 26 Well-quasi-orderability

There exists Z ∈ MX (p) such that Z ⊆ Y.

In other words, if X is wqo, then unboundedness in X of any parameter can be characterised in terms of minimal classes.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 7 / 26 the smallest class of graphs containing a one-vertex graph, and closed under disjoint union and join. the class of graphs, all of whose induced subgraphs are either disconnected, or the complement of a disconnected graph.

the class of graphs that avoid P4 as an induced subgraph. the class of graphs of clique-width at most 2.

The class of cographs, or complement-reducible graphs, has been studied extensively and has many known characterisations. It is:

Theorem (Damaschke) The class of cographs is wqo under the induced subgraph relation. P. Damaschke, Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990).

Cographs

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 the smallest class of graphs containing a one-vertex graph, and closed under disjoint union and join. the class of graphs, all of whose induced subgraphs are either disconnected, or the complement of a disconnected graph.

the class of graphs that avoid P4 as an induced subgraph. the class of graphs of clique-width at most 2.

It is:

Theorem (Damaschke) The class of cographs is wqo under the induced subgraph relation. P. Damaschke, Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990).

Cographs

The class of cographs, or complement-reducible graphs, has been studied extensively and has many known characterisations.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 the class of graphs, all of whose induced subgraphs are either disconnected, or the complement of a disconnected graph.

the class of graphs that avoid P4 as an induced subgraph. the class of graphs of clique-width at most 2. Theorem (Damaschke) The class of cographs is wqo under the induced subgraph relation. P. Damaschke, Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990).

Cographs

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 the class of graphs that avoid P4 as an induced subgraph. the class of graphs of clique-width at most 2. Theorem (Damaschke) The class of cographs is wqo under the induced subgraph relation. P. Damaschke, Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990).

Cographs

The class of cographs, or complement-reducible graphs, has been studied extensively and has many known characterisations. It is: the smallest class of graphs containing a one-vertex graph, and closed under disjoint union and join. the class of graphs, all of whose induced subgraphs are either disconnected, or the complement of a disconnected graph.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 the class of graphs of clique-width at most 2. Theorem (Damaschke) The class of cographs is wqo under the induced subgraph relation. P. Damaschke, Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990).

Cographs

the class of graphs that avoid P4 as an induced subgraph.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 Theorem (Damaschke) The class of cographs is wqo under the induced subgraph relation. P. Damaschke, Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990).

Cographs

the class of graphs that avoid P4 as an induced subgraph. the class of graphs of clique-width at most 2.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 Cographs

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 8 / 26 One example is the behaviour of linear clique-width. This parameter is unbounded in the class of cographs. F. Gurski, E. Wanke, On the relationship between NLC-width and linear NLC-width. Theoretical Computer Science 347, 76–89 (2005). A class of cographs has unbounded linear clique-width if and only if it contains all quasi-threshold graphs or their complements. R. Brignall, N. Korpelainen, V. Vatter, Linear clique-width for hereditary classes of cographs. J. Graph Theory 84, 501-511 (2017).

We present several more results of this type: for various parameters that are unbounded in the class of cographs, we ﬁnd the minimal classes where the parameters are unbounded.

Cographs

Cographs provide a “safe” environment, in which many interesting things still happen.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 9 / 26 This parameter is unbounded in the class of cographs. F. Gurski, E. Wanke, On the relationship between NLC-width and linear NLC-width. Theoretical Computer Science 347, 76–89 (2005). A class of cographs has unbounded linear clique-width if and only if it contains all quasi-threshold graphs or their complements. R. Brignall, N. Korpelainen, V. Vatter, Linear clique-width for hereditary classes of cographs. J. Graph Theory 84, 501-511 (2017).

We present several more results of this type: for various parameters that are unbounded in the class of cographs, we ﬁnd the minimal classes where the parameters are unbounded.

Cographs

Cographs provide a “safe” environment, in which many interesting things still happen. One example is the behaviour of linear clique-width.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 9 / 26 A class of cographs has unbounded linear clique-width if and only if it contains all quasi-threshold graphs or their complements. R. Brignall, N. Korpelainen, V. Vatter, Linear clique-width for hereditary classes of cographs. J. Graph Theory 84, 501-511 (2017).

We present several more results of this type: for various parameters that are unbounded in the class of cographs, we ﬁnd the minimal classes where the parameters are unbounded.

Cographs

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 9 / 26 We present several more results of this type: for various parameters that are unbounded in the class of cographs, we ﬁnd the minimal classes where the parameters are unbounded.

Cographs

Cographs provide a “safe” environment, in which many interesting things still happen. One example is the behaviour of linear clique-width. This parameter is unbounded in the class of cographs. F. Gurski, E. Wanke, On the relationship between NLC-width and linear NLC-width. Theoretical Computer Science 347, 76–89 (2005). A class of cographs has unbounded linear clique-width if and only if it contains all quasi-threshold graphs or their complements. R. Brignall, N. Korpelainen, V. Vatter, Linear clique-width for hereditary classes of cographs. J. Graph Theory 84, 501-511 (2017).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 9 / 26 Cographs

We present several more results of this type: for various parameters that are unbounded in the class of cographs, we ﬁnd the minimal classes where the parameters are unbounded.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 9 / 26 Deﬁnition The co-chromatic number of G is the minimum number of subsets in a partition of V (G) such that any subset is a clique or an independent set.

Theorem Let U be the class of unions of cliques. U and U are the only minimal hereditary subclasses of cographs of unbounded cochromatic number.

Co-chromatic number

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 10 / 26 Theorem Let U be the class of unions of cliques. U and U are the only minimal hereditary subclasses of cographs of unbounded cochromatic number.

Co-chromatic number

Deﬁnition The co-chromatic number of G is the minimum number of subsets in a partition of V (G) such that any subset is a clique or an independent set.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 10 / 26 Co-chromatic number

Deﬁnition The co-chromatic number of G is the minimum number of subsets in a partition of V (G) such that any subset is a clique or an independent set.

Theorem Let U be the class of unions of cliques. U and U are the only minimal hereditary subclasses of cographs of unbounded cochromatic number.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 10 / 26 Lettericity is a parameter of interest in the study of well-quasi-orderability: classes of bounded lettericity are wqo.

It is also related to the notion of geometric griddability in the study of permutations.

Theorem Let M be the class of graphs of vertex degree at most 1. M and M are the only minimal hereditary subclasses of cographs of unbounded lettericity.

Lettericity (Petkovˇsek,2002)

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 11 / 26 It is also related to the notion of geometric griddability in the study of permutations.

Theorem Let M be the class of graphs of vertex degree at most 1. M and M are the only minimal hereditary subclasses of cographs of unbounded lettericity.

Lettericity (Petkovˇsek,2002)

Lettericity is a parameter of interest in the study of well-quasi-orderability: classes of bounded lettericity are wqo.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 11 / 26 Theorem Let M be the class of graphs of vertex degree at most 1. M and M are the only minimal hereditary subclasses of cographs of unbounded lettericity.

Lettericity (Petkovˇsek,2002)

Lettericity is a parameter of interest in the study of well-quasi-orderability: classes of bounded lettericity are wqo.

It is also related to the notion of geometric griddability in the study of permutations.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 11 / 26 Lettericity (Petkovˇsek,2002)

Lettericity is a parameter of interest in the study of well-quasi-orderability: classes of bounded lettericity are wqo.

It is also related to the notion of geometric griddability in the study of permutations.

Theorem Let M be the class of graphs of vertex degree at most 1. M and M are the only minimal hereditary subclasses of cographs of unbounded lettericity.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 11 / 26 Theorem M is the only minimal hereditary subclass of cographs of unbounded boxicity.

Boxicity

Deﬁnition The boxicity of a graph G is the minimum dimension in which G can be represented as an intersection graph of hyper-rectangles.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 12 / 26 Boxicity

Deﬁnition The boxicity of a graph G is the minimum dimension in which G can be represented as an intersection graph of hyper-rectangles.

Theorem M is the only minimal hereditary subclass of cographs of unbounded boxicity.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 12 / 26 Theorem Let B be the class of complete bipartite graphs, and F the class of star forests. K, B and F are the only minimal hereditary subclasses of cographs of unbounded H-index.

H-index

Deﬁnition The H-index of a graph G is the largest k ≥ 0 such that G has k vertices of degree at least k.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 13 / 26 H-index

Deﬁnition The H-index of a graph G is the largest k ≥ 0 such that G has k vertices of degree at least k.

Theorem Let B be the class of complete bipartite graphs, and F the class of star forests. K, B and F are the only minimal hereditary subclasses of cographs of unbounded H-index.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 13 / 26 Theorem K and M are the only minimal hereditary subclasses of cographs of unbounded achromatic number.

Achromatic number

Deﬁnition A complete k-colouring of a graph G is a partition of G into k independent sets such that any two of the independent sets have an edge between them. The achromatic number of G is the maximum k such that G admits a complete k-colouring.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 14 / 26 Achromatic number

Theorem K and M are the only minimal hereditary subclasses of cographs of unbounded achromatic number.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 14 / 26 K1 ∈ Q;

If G1, G2 ∈ Q, then G1 ∪ G2 ∈ Q; If G ∈ Q, then v × G ∈ Q.

Deﬁnition The class Q of quasi-threshold graphs is the smallest subclass of cographs such that:

Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Contiguity

Deﬁnition The contiguity of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 15 / 26 K1 ∈ Q;

If G1, G2 ∈ Q, then G1 ∪ G2 ∈ Q; If G ∈ Q, then v × G ∈ Q.

Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Contiguity

Deﬁnition The contiguity of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

Deﬁnition The class Q of quasi-threshold graphs is the smallest subclass of cographs such that:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 15 / 26 If G1, G2 ∈ Q, then G1 ∪ G2 ∈ Q; If G ∈ Q, then v × G ∈ Q.

Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Contiguity

Deﬁnition The contiguity of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

Deﬁnition The class Q of quasi-threshold graphs is the smallest subclass of cographs such that:

K1 ∈ Q;

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 15 / 26 If G ∈ Q, then v × G ∈ Q.

Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Contiguity

Deﬁnition The contiguity of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

Deﬁnition The class Q of quasi-threshold graphs is the smallest subclass of cographs such that:

K1 ∈ Q;

If G1, G2 ∈ Q, then G1 ∪ G2 ∈ Q;

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 15 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Contiguity

Deﬁnition The contiguity of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

Deﬁnition The class Q of quasi-threshold graphs is the smallest subclass of cographs such that:

K1 ∈ Q;

If G1, G2 ∈ Q, then G1 ∪ G2 ∈ Q; If G ∈ Q, then v × G ∈ Q.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 15 / 26 Contiguity

Deﬁnition The contiguity of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

Deﬁnition The class Q of quasi-threshold graphs is the smallest subclass of cographs such that:

K1 ∈ Q;

If G1, G2 ∈ Q, then G1 ∪ G2 ∈ Q; If G ∈ Q, then v × G ∈ Q.

Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 15 / 26 M. Jean, An interval graph is a comparability graph. J. Combinatorial Theory 7(2), 189–190 (1969).

P. C. Fishburn, An interval graph is not a comparability graph. J. Combinatorial Theory 8(4), 442–443 (1970).

For classes of graphs, the minimal forbidden induced subgraph characterisation can be very helpful.

It can be used to compare classes of graphs: Proposition Free(S) ⊆ Free(T ) if and only if every graph H ∈ T has a graph G ∈ S with G ≤ H.

A hierarchy of parameters

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 16 / 26 M. Jean, An interval graph is a comparability graph. J. Combinatorial Theory 7(2), 189–190 (1969).

P. C. Fishburn, An interval graph is not a comparability graph. J. Combinatorial Theory 8(4), 442–443 (1970). It can be used to compare classes of graphs: Proposition Free(S) ⊆ Free(T ) if and only if every graph H ∈ T has a graph G ∈ S with G ≤ H.

A hierarchy of parameters

For classes of graphs, the minimal forbidden induced subgraph characterisation can be very helpful.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 16 / 26 P. C. Fishburn, An interval graph is not a comparability graph. J. Combinatorial Theory 8(4), 442–443 (1970). It can be used to compare classes of graphs: Proposition Free(S) ⊆ Free(T ) if and only if every graph H ∈ T has a graph G ∈ S with G ≤ H.

A hierarchy of parameters

For classes of graphs, the minimal forbidden induced subgraph characterisation can be very helpful.

M. Jean, An interval graph is a comparability graph. J. Combinatorial Theory 7(2), 189–190 (1969).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 16 / 26 It can be used to compare classes of graphs: Proposition Free(S) ⊆ Free(T ) if and only if every graph H ∈ T has a graph G ∈ S with G ≤ H.

A hierarchy of parameters

For classes of graphs, the minimal forbidden induced subgraph characterisation can be very helpful.

M. Jean, An interval graph is a comparability graph. J. Combinatorial Theory 7(2), 189–190 (1969).

P. C. Fishburn, An interval graph is not a comparability graph. J. Combinatorial Theory 8(4), 442–443 (1970).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 16 / 26 Proposition Free(S) ⊆ Free(T ) if and only if every graph H ∈ T has a graph G ∈ S with G ≤ H.

A hierarchy of parameters

For classes of graphs, the minimal forbidden induced subgraph characterisation can be very helpful.

M. Jean, An interval graph is a comparability graph. J. Combinatorial Theory 7(2), 189–190 (1969).

P. C. Fishburn, An interval graph is not a comparability graph. J. Combinatorial Theory 8(4), 442–443 (1970). It can be used to compare classes of graphs:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 16 / 26 A hierarchy of parameters

For classes of graphs, the minimal forbidden induced subgraph characterisation can be very helpful.

M. Jean, An interval graph is a comparability graph. J. Combinatorial Theory 7(2), 189–190 (1969).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 16 / 26 Deﬁnition Let p and q be two parameters. We say p is stronger than q in the universe of cographs if boundedness of q implies boundedness of p in a class of cographs.

Example H-index is stronger than maximum degree.

Proposition Write M(p), M(q) for the respective sets of minimal classes of cographs where p and q are unbounded. p is stronger than q if and only if for every class Y ∈ M(q) there is a class X ∈ M(p) with X ⊆ Y.

A hierarchy of parameters

Analogously, we can compare parameters.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 17 / 26 Example H-index is stronger than maximum degree.

A hierarchy of parameters

Analogously, we can compare parameters. Deﬁnition Let p and q be two parameters. We say p is stronger than q in the universe of cographs if boundedness of q implies boundedness of p in a class of cographs.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 17 / 26 Proposition Write M(p), M(q) for the respective sets of minimal classes of cographs where p and q are unbounded. p is stronger than q if and only if for every class Y ∈ M(q) there is a class X ∈ M(p) with X ⊆ Y.

A hierarchy of parameters

Example H-index is stronger than maximum degree.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 17 / 26 A hierarchy of parameters

Example H-index is stronger than maximum degree.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 17 / 26 A hierarchy of parameters

linear clique-width, contiguity (Q, Q)

co-chromatic number boxicity (U, U) (M)

chromatic number lettericity (K) (M, M)

tree-width, degeneracy neighbourhood diversity (B, K) (M, M, T )

H-index achromatic number (F, B, K) (M, K)

maximum degree matching number (S, K) (M, B, K)

Figure: A Hasse diagram of graph parameters within the universe of cographs

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 18 / 26 There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Lemma For a class X , the following are equivalent:

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

There exists a parameter pX for which X is minimal.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Proof: “ =⇒ ”

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 Hence H is induced in every Gi for i large enough.

Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 Some further questions

In the hierarchy, there are 13 parameters, but only 11 diﬀerent classes (including complements). Why is that?

Lemma For a class X , the following are equivalent:

There exists a parameter pX for which X is minimal.

There exists a universal sequence G1, G2,... of graphs in X such that Gi contains any graph in X on i vertices.

Proof: “ =⇒ ” Since pX is unbounded in X , there exists a sequence G1, G2,... with pX (Gi ) → ∞.

Since X is minimal, for any graph H ∈ X , X ∩ Free(H) is of bounded pX .

Hence H is induced in every Gi for i large enough.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 19 / 26 We get a universal subsequence Gt1 , Gt2 ,... .

“ ⇐= ” Suppose X has a universal sequence G1, G2,... .

Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

This helps, but it is not a complete answer.

Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 “ ⇐= ” Suppose X has a universal sequence G1, G2,... .

Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

This helps, but it is not a complete answer.

Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

We get a universal subsequence Gt1 , Gt2 ,... .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 Suppose X has a universal sequence G1, G2,... .

Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

This helps, but it is not a complete answer.

Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

We get a universal subsequence Gt1 , Gt2 ,... .

“ ⇐= ”

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

This helps, but it is not a complete answer.

Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

We get a universal subsequence Gt1 , Gt2 ,... .

“ ⇐= ” Suppose X has a universal sequence G1, G2,... .

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

This helps, but it is not a complete answer.

Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

We get a universal subsequence Gt1 , Gt2 ,... .

“ ⇐= ” Suppose X has a universal sequence G1, G2,... .

Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 This helps, but it is not a complete answer.

Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

We get a universal subsequence Gt1 , Gt2 ,... .

“ ⇐= ” Suppose X has a universal sequence G1, G2,... .

Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 Some further questions

Since there are ﬁnitely many graphs on i vertices, all of them belong to a common Gi .

We get a universal subsequence Gt1 , Gt2 ,... .

“ ⇐= ” Suppose X has a universal sequence G1, G2,... .

Deﬁne pX (G) as the largest i such that G contains every graph in X on i vertices as an induced subgraph.

Then pX is unbounded in X (because of the sequence), and unbounded in every proper subclass by construction.

This helps, but it is not a complete answer.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 20 / 26 H. Bodlaender, Achromatic number is NP–complete for cographs and interval graphs. Information Processing Letters 31(3), 135–138 (1989). P. Damaschke, Induced subgraph isomorphism for cographs is NP–complete. In: WG’90, LNCS 487, 72–78 (1991).

More generally, are there conditions under which we can predict the behaviour of parameters in the class of all graphs based on their behaviour in a restricted setting (e.g., wqo classes)?

A similar study could be carried regarding hardness of algorithms.

Finally, note that the sets M(p) we found so far are all ﬁnite. Is this the case for all parameters?

H-index, for instance, is characterised in the class of all graphs by the same minimal classes. When does this happen?

Some further questions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 H. Bodlaender, Achromatic number is NP–complete for cographs and interval graphs. Information Processing Letters 31(3), 135–138 (1989). P. Damaschke, Induced subgraph isomorphism for cographs is NP–complete. In: WG’90, LNCS 487, 72–78 (1991).

A similar study could be carried regarding hardness of algorithms.

Finally, note that the sets M(p) we found so far are all ﬁnite. Is this the case for all parameters?

More generally, are there conditions under which we can predict the behaviour of parameters in the class of all graphs based on their behaviour in a restricted setting (e.g., wqo classes)?

Some further questions

H-index, for instance, is characterised in the class of all graphs by the same minimal classes. When does this happen?

Is this the case for all parameters?

A similar study could be carried regarding hardness of algorithms.

Finally, note that the sets M(p) we found so far are all ﬁnite.

Some further questions

H-index, for instance, is characterised in the class of all graphs by the same minimal classes. When does this happen? More generally, are there conditions under which we can predict the behaviour of parameters in the class of all graphs based on their behaviour in a restricted setting (e.g., wqo classes)?

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 Is this the case for all parameters?

H. Bodlaender, Achromatic number is NP–complete for cographs and interval graphs. Information Processing Letters 31(3), 135–138 (1989). P. Damaschke, Induced subgraph isomorphism for cographs is NP–complete. In: WG’90, LNCS 487, 72–78 (1991).

Finally, note that the sets M(p) we found so far are all ﬁnite.

Some further questions

A similar study could be carried regarding hardness of algorithms.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 Is this the case for all parameters?

P. Damaschke, Induced subgraph isomorphism for cographs is NP–complete. In: WG’90, LNCS 487, 72–78 (1991).

Finally, note that the sets M(p) we found so far are all ﬁnite.

Some further questions

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 Is this the case for all parameters?

Finally, note that the sets M(p) we found so far are all ﬁnite.

Some further questions

A similar study could be carried regarding hardness of algorithms. H. Bodlaender, Achromatic number is NP–complete for cographs and interval graphs. Information Processing Letters 31(3), 135–138 (1989). P. Damaschke, Induced subgraph isomorphism for cographs is NP–complete. In: WG’90, LNCS 487, 72–78 (1991).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 Is this the case for all parameters?

Some further questions

Finally, note that the sets M(p) we found so far are all ﬁnite.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 Some further questions

Finally, note that the sets M(p) we found so far are all ﬁnite. Is this the case for all parameters?

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 21 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 Bonus: Minimal classes of unbounded contiguity

Deﬁnition The contiguity cont(G) of a graph G is the minimum number k such that V (G) admits a linear ordering in which the neighbourhood of each vertex consists of at most k intervals.

Theorem Q and Q are the only minimal hereditary subclasses of cographs of unbounded contiguity.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 22 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

Bonus: Minimal classes of unbounded contiguity

Proof.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 v u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

L1 L2 L3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u1 u2 u3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

u2 u3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

u1 u2

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3 u2

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v L

u1 u2 u3

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u3 u2

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

u1 v L

u1 u2 u3

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

u1 v u2 L

u1 u2 u3

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

u1 v u3 u2 L

u1 u2 u3

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

u1 v u3 u2 L

u1 u2 u3

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 u1 u3

u1 u2 u3

Bonus: Minimal classes of unbounded contiguity

Proof. Step 1: Q and Q have unbounded contiguity. Remark cont(G) ≤ cont(G) + 1, so it suﬃces to prove it for Q.

Lemma Suppose cont(G) = k. Then cont (v × (G ∪ G ∪ G)) > k.

v u2 L

L1 L2 L3

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 23 / 26 0 If G = G1 × · · · × Gp, each Gp is K -free 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2. If G = G1 ∪ · · · ∪ Gp, each Gi is connected 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 0 If G = G1 × · · · × Gp, each Gp is K -free 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2. If G = G1 ∪ · · · ∪ Gp, each Gi is connected 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof:

Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|.

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

Case 1: K = K 0 × w (or H = H0 ∪ v).

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2. If G = G1 ∪ · · · ∪ Gp, each Gi is connected 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

0 If G = G1 × · · · × Gp, each Gp is K -free

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 If G = G1 ∪ · · · ∪ Gp, each Gi is connected 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2.

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

0 If G = G1 × · · · × Gp, each Gp is K -free

=⇒ cont(G) ≤ c(H, K 0) + 2.

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

0 If G = G1 × · · · × Gp, each Gp is K -free 0 =⇒ cont(Gi ) ≤ c(H, K )

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

If G = G1 ∪ · · · ∪ Gp, each Gi is connected

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

0 If G = G1 × · · · × Gp, each Gp is K -free 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

0 If G = G1 × · · · × Gp, each Gp is K -free 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2. If G = G1 ∪ · · · ∪ Gp, each Gi is connected

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 =⇒ cont(G) ≤ c(H, K 0) + 2.

Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

0 If G = G1 × · · · × Gp, each Gp is K -free 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2. If G = G1 ∪ · · · ∪ Gp, each Gi is connected 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 Bonus: Minimal classes of unbounded contiguity

Lemma Let H ∈ Q, K ∈ Q. There exists a constant c(H, K) such that the contiguity of (H, K)-free cographs is at most c(H, K).

Proof: By induction on |V (H)| + |V (K)|. Remark

If G = G1 ∪ · · · ∪ Gp, then cont(G) = maxi cont(Gi ).

If G = G1 × · · · × Gp, then cont(G) ≤ maxi cont(Gi ) + 2.

Case 1: K = K 0 × w (or H = H0 ∪ v).

0 If G = G1 × · · · × Gp, each Gp is K -free 0 0 =⇒ cont(Gi ) ≤ c(H, K ) =⇒ cont(G) ≤ c(H, K ) + 2. If G = G1 ∪ · · · ∪ Gp, each Gi is connected 0 0 =⇒ cont(Gi ) ≤ c(H, K ) + 2 =⇒ cont(G) ≤ c(H, K ) + 2.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 24 / 26 +1

0 G1 G1

0 G2 G2

0 Gk−1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 +1

0 G1 G1

0 G2 G2

0 Gk−1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 +1

0 G1 G1

0 G2 G2

0 Gk−1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 +1

0 G2 G2

0 Gk−1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 +1

0 Gk−1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 +1

0 Gk−1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 +1

0 Gk Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

0 Gk−1

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

0 Gk−1

0 Gk Gk

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

0 Gk−1

Gk+1 Gk

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 L := L1L3.

Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

0 Gk−1

Gk+1 Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

0 Gk−1

Gk+1 Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3L5 ... Lk−1Lk+1Lk Lk−2 ... L6L4L2.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 Bonus: Minimal classes of unbounded contiguity

Case 2: H = H0 ∪ H00 and K = K 0 × K 00.

0 G0

0 G1 G1

0 G2 G2

0 Gk−1

Gk+1 Gk

0 G = G1 ∪ (G2 × (G3 ∪ ... (Gk × Gk+1))) with cont(Gi ) ≤ c (H, K).

L := L1L3L5 ... Lk−1Lk+1Lk Lk−2 ... L6L4L2.

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 25 / 26 Questions?

Thank you!

B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 26 / 26