Boundary properties of factorial classes of graphs
Victor Zamaraev
Laboratory of Algorithms and Technologies for Networks ...
Boundary properties of factorial classes of graphs
Introduction
2 / 28
Boundary properties of factorial classes of graphs
Introduction
All considered graphs are simple (undirected, without loop...
Boundary properties of factorial classes of graphs
Introduction
All considered graphs are simple (undirected, without loop...
Boundary properties of factorial classes of graphs
Introduction
Definition
A class is a set of graphs closed under isomorph...
Boundary properties of factorial classes of graphs
Introduction
Definition
A class is a set of graphs closed under isomorph...
Boundary properties of factorial classes of graphs
Introduction
Definition
A class is a set of graphs closed under isomorph...
Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
f...
Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
f...
Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
f...
Boundary properties of factorial classes of graphs
Introduction
Every hereditary graph class X can be defined by a set of
f...
Boundary properties of factorial classes of graphs
Introduction
For a class X denote by Xn the set of n-vertex graphs from...
Boundary properties of factorial classes of graphs
Introduction
For a class X denote by Xn the set of n-vertex graphs from...
Boundary properties of factorial classes of graphs
Introduction
For a class X denote by Xn the set of n-vertex graphs from...
Boundary properties of factorial classes of graphs
Introduction
Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A...
Boundary properties of factorial classes of graphs
Introduction
Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A...
Boundary properties of factorial classes of graphs
Introduction
Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Let c(X) = 1
Question
What are possible rates of growth of...
Boundary properties of factorial classes of graphs
Introduction
Constant
Polynomial
Exponential
Factorial layer
Classes wi...
Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
10 / 28
Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
Polynomi...
Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
Polynomi...
Boundary properties of factorial classes of graphs
Introduction
Example
Constant class: Co – complete graphs (1).
Polynomi...
Boundary properties of factorial classes of graphs
Introduction
Alekseev V.E. (1997)
Constant classes: log2 |Xn| = Θ(1).
P...
Boundary properties of factorial classes of graphs
Introduction
Alekseev V.E. (1997)
Constant classes: log2 |Xn| = Θ(1).
P...
Boundary properties of factorial classes of graphs
Introduction
Constant
Polynomial
Exponential
Factorial layer
Classes wi...
Boundary properties of factorial classes of graphs
Introduction
Balogh J., Bollob´as B., Weinreich D. (2000)
Constant clas...
Boundary properties of factorial classes of graphs
Introduction
Balogh J., Bollob´as B., Weinreich D. (2000)
Constant clas...
Boundary properties of factorial classes of graphs
Introduction
Examples of factorial classes:
forests
planar graphs
line ...
Boundary properties of factorial classes of graphs
Introduction
Problem
Characterize factorial layer.
15 / 28
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
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Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Constant
...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Constant
...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn|...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn|...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn|...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
log2 |Xn|...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Theorem (...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Theorem (...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Minimal superfactorial classes
Denote by...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Limit classes
Definition
Given a sequence...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Limit classes
Definition
Given a sequence...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Limit classes
Definition
Given a sequence...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Boundary classes
Definition
A limit class...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Boundary classes
Definition
A limit class...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Boundary classes
Definition
A limit class...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There a...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There a...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There a...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Are there more boundary classes?
There a...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Lozin’s conjecture
Conjecture (Lozin’s c...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4)...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4)...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4)...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4)...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
B2 = Free(C4)...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
Theorem
There...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Proper boundary subclasses
Theorem
There...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Open problems
Open question
Find a minim...
Boundary properties of factorial classes of graphs
Minimal superfactorial classes
Open problems
Open question
Find a minim...
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Victor Zamaraev – Boundary properties of factorial classes of graphs

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For a class of graphs X, let X_n be the number of graphs with vertex set {1,...,n} in the class X, also known as the speed of X. It is known that in the family of hereditary classes (i.e. those that are closed under taking induced subgraphs) the speeds constitute discrete layers and the first four lower layers are constant, polynomial, exponential, and factorial. For each of these four layers a complete list of minimal classes is available, and this information allows to provide a global structural characterization for the first three of them. The minimal layer for which no such characterization is known is the factorial one. A possible approach to obtaining such a characterization could be through identifying all minimal superfactorial classes. However, no such class is known and possibly no such class exists. To overcome this difficulty, we employ the notion of boundary classes that has been recently introduced to study algorithmic graph problems and reveal the first few boundary classes for the factorial layer.

Joint work with Vadim Lozin.

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Victor Zamaraev – Boundary properties of factorial classes of graphs

  1. 1. Boundary properties of factorial classes of graphs Victor Zamaraev Laboratory of Algorithms and Technologies for Networks Analysis (LATNA), Higher School of Economics Joint work with Vadim Lozin, University of Warwick Workshop on Extremal Graph Theory 6 June 2014
  2. 2. Boundary properties of factorial classes of graphs Introduction 2 / 28
  3. 3. Boundary properties of factorial classes of graphs Introduction All considered graphs are simple (undirected, without loops and without multiple edges). 3 / 28
  4. 4. Boundary properties of factorial classes of graphs Introduction All considered graphs are simple (undirected, without loops and without multiple edges). Graphs are labeled by natural numbers 1, . . . , n 6 4 5 1 2 3 3 / 28
  5. 5. Boundary properties of factorial classes of graphs Introduction Definition A class is a set of graphs closed under isomorphism. 4 / 28
  6. 6. Boundary properties of factorial classes of graphs 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 induced subgraphs. 4 / 28
  7. 7. Boundary properties of factorial classes of graphs 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 induced subgraphs. Exapmle Let X be a hereditary class and W4 ∈ X. Then C4 ∈ X. 1 2 3 4 5 1 2 3 4 W4 C4 4 / 28
  8. 8. Boundary properties of factorial classes of graphs Introduction Every hereditary graph class X can be defined by a set of forbidden induced subgraphs. 5 / 28
  9. 9. Boundary properties of factorial classes of graphs Introduction Every hereditary graph class X can be defined by a set of forbidden induced subgraphs. Let M be a set of graphs. Then Free(M) denotes the set of all graphs not containing induced subgraphs isomorphic to graphs from M. 5 / 28
  10. 10. Boundary properties of factorial classes of graphs Introduction Every hereditary graph class X can be defined by a set of forbidden induced subgraphs. Let M be a set of graphs. Then Free(M) denotes the set of all graphs not containing induced subgraphs isomorphic to graphs from M. Statement Class X is hereditary if and only if there exists M such that X = Free(M). We say that graphs in X are M-free. 5 / 28
  11. 11. Boundary properties of factorial classes of graphs Introduction Every hereditary graph class X can be defined by a set of forbidden induced subgraphs. Let M be a set of graphs. Then Free(M) denotes the set of all graphs not containing induced subgraphs isomorphic to graphs from M. Statement Class X is hereditary if and only if there exists M such that X = Free(M). We say that graphs in X are M-free. Example For the class of bipartite graphs M is {C3, C5, C7, . . . }, i.e. B = Free(C3, C5, C7, . . . ). 5 / 28
  12. 12. Boundary properties of factorial classes of graphs Introduction For a class X denote by Xn the set of n-vertex graphs from X. 6 / 28
  13. 13. Boundary properties of factorial classes of graphs 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(n 2) = 2n(n−1)/2 6 / 28
  14. 14. Boundary properties of factorial classes of graphs 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(n 2) = 2n(n−1)/2 log2 |Pn| = Θ(n2) 6 / 28
  15. 15. Boundary properties of factorial classes of graphs Introduction Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A., 1994) For every infinite hereditary class X, which is not the class of all graphs: log2 |Xn| = 1 − 1 c(X) n2 2 + o(n2 ), (1) where c(X) ∈ N is the index of class X. 7 / 28
  16. 16. Boundary properties of factorial classes of graphs Introduction Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A., 1994) For every infinite hereditary class X, which is not the class of all graphs: log2 |Xn| = 1 − 1 c(X) n2 2 + o(n2 ), (1) where c(X) ∈ N is the index of class X. (i) For c(X) > 1, log2 |Xn| = Θ(n2) 7 / 28
  17. 17. Boundary properties of factorial classes of graphs Introduction Theorem (Alekseev V. E., 1992; Bollob´as B. and Thomason A., 1994) For every infinite hereditary class X, which is not the class of all graphs: log2 |Xn| = 1 − 1 c(X) n2 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) 7 / 28
  18. 18. Boundary properties of factorial classes of graphs Introduction Let c(X) = 1 Question What are possible rates of growth of a function log2 |Xn|? 8 / 28
  19. 19. Boundary properties of factorial classes of graphs 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). 8 / 28
  20. 20. Boundary properties of factorial classes of graphs 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). 8 / 28
  21. 21. Boundary properties of factorial classes of graphs 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). 8 / 28
  22. 22. Boundary properties of factorial classes of graphs 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). 8 / 28
  23. 23. Boundary properties of factorial classes of graphs 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. 8 / 28
  24. 24. Boundary properties of factorial classes of graphs 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 with log2 |Xn| = Θ( √ n). 8 / 28
  25. 25. Boundary properties of factorial classes of graphs Introduction Constant Polynomial Exponential Factorial layer Classes with index 1 9 / 28
  26. 26. Boundary properties of factorial classes of graphs Introduction Example Constant class: Co – complete graphs (1). 10 / 28
  27. 27. Boundary properties of factorial classes of graphs Introduction Example Constant class: Co – complete graphs (1). Polynomial class: E1 – graphs with at most one edge ( n 2 + 1). 10 / 28
  28. 28. Boundary properties of factorial classes of graphs Introduction Example Constant class: Co – complete graphs (1). Polynomial class: E1 – graphs with at most one edge ( n 2 + 1). Exponential class: Co + Co (2n−1). 10 / 28
  29. 29. Boundary properties of factorial classes of graphs Introduction Example Constant class: Co – complete graphs (1). Polynomial class: E1 – graphs with at most one edge ( n 2 + 1). Exponential class: Co + Co (2n−1). Factorial class: F – forests (nn−2 < |Fn| < n2n). 10 / 28
  30. 30. Boundary properties of factorial classes of graphs Introduction Alekseev V.E. (1997) 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. 11 / 28
  31. 31. Boundary properties of factorial classes of graphs Introduction Alekseev V.E. (1997) 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. 1 Structural characterizations were obtained for the first three layers. 2 In every of the four layers all minimal classes were found. 11 / 28
  32. 32. Boundary properties of factorial classes of graphs Introduction Constant Polynomial Exponential Factorial layer Classes with index 1 12 / 28
  33. 33. Boundary properties of factorial classes of graphs Introduction Balogh J., Bollob´as B., Weinreich D. (2000) 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. 13 / 28
  34. 34. Boundary properties of factorial classes of graphs Introduction Balogh J., Bollob´as B., Weinreich D. (2000) 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. In addition 1 Characterized lower part of the factorial layer, i.e. classes with |Xn| < n(1+o(1))n. 13 / 28
  35. 35. Boundary properties of factorial classes of graphs Introduction Examples of factorial classes: forests planar graphs line graphs cographs permutation graphs threshold graphs graphs of bounded vertex degree graphs of bounded clique-width et al. 14 / 28
  36. 36. Boundary properties of factorial classes of graphs Introduction Problem Characterize factorial layer. 15 / 28
  37. 37. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 16 / 28
  38. 38. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes Constant Polynomial Exponential Factorial Classes with index 1
  39. 39. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes Constant Polynomial Exponential Factorial Classes with index 1 ? ? ? 17 / 28
  40. 40. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes log2 |Xn| = Θ(n log2 n) 18 / 28
  41. 41. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes log2 |Xn| = Θ(n log2 n) CB = Free(C3, C5, C6, C7, . . .) 18 / 28
  42. 42. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes log2 |Xn| = Θ(n log2 n) CB = Free(C3, C5, C6, C7, . . .) Theorem (Spinrad J. P., 1995) log2 |CBn| = Θ(n log2 n) 18 / 28
  43. 43. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 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? 18 / 28
  44. 44. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 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 19 / 28
  45. 45. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 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? 19 / 28
  46. 46. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes Denote by Bk the class of (C4, C6, ..., C2k)-free bipartite graphs. 20 / 28
  47. 47. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 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. 20 / 28
  48. 48. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 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 . . . . 20 / 28
  49. 49. Boundary properties of factorial classes of graphs Minimal superfactorial classes Minimal superfactorial classes 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. 20 / 28
  50. 50. Boundary properties of factorial classes of graphs Minimal superfactorial classes Limit classes Definition Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we will say that the sequence converges to a class X if i≥1 Xi = X. 21 / 28
  51. 51. Boundary properties of factorial classes of graphs Minimal superfactorial classes Limit classes Definition Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we will say that the sequence converges to a class X if i≥1 Xi = X. Example The sequence B2 ⊃ B3 ⊃ B4 . . . converges to the factorial class F of forests, i.e. i≥1 Bi = F. 21 / 28
  52. 52. Boundary properties of factorial classes of graphs Minimal superfactorial classes Limit classes Definition Given a sequence X1 ⊇ X2 ⊇ X3 ⊇ . . . of graph classes, we will say that the sequence converges to a class X if i≥1 Xi = X. Example The sequence B2 ⊃ B3 ⊃ B4 . . . converges to the factorial class F of forests, i.e. i≥1 Bi = F. Definition A class X of graphs is a limit class (for the factorial layer) if there is a sequence of superfactorial classes converging to X. 21 / 28
  53. 53. Boundary properties of factorial classes of graphs Minimal superfactorial classes Boundary classes Definition A limit class is called boundary (or minimal) if it does not properly contain any other limit class. 22 / 28
  54. 54. Boundary properties of factorial classes of graphs Minimal superfactorial classes Boundary classes Definition A limit class is called boundary (or minimal) if it does not properly contain any other limit class. Theorem A finitely defined class is superfactorial if and only if it contains a boundary class. 22 / 28
  55. 55. Boundary properties of factorial classes of graphs Minimal superfactorial classes Boundary classes Definition A limit class is called boundary (or minimal) if it does not properly contain any other limit class. Theorem A finitely defined class is superfactorial if and only if it contains a boundary class. Theorem The class of forests is a boundary class. 22 / 28
  56. 56. Boundary properties of factorial classes of graphs Minimal superfactorial classes Are there more boundary classes? There are five more boundary classes, which can be easly obtained from the class of forests. 23 / 28
  57. 57. Boundary properties of factorial classes of graphs Minimal superfactorial classes Are there more boundary classes? There are five more boundary classes, which can be easly obtained from the class of forests. Two of them are: 1 complements of forests; 2 bipartite complements of forests; 23 / 28
  58. 58. Boundary properties of factorial classes of graphs Minimal superfactorial classes Are there more boundary classes? There are five more boundary classes, which can be easly obtained from the class of forests. Two of them are: 1 complements of forests; 2 bipartite complements of forests; 1 5 2 6 3 7 4 8 F 1 5 2 6 3 7 4 8 Bipartite complement of F 23 / 28
  59. 59. Boundary properties of factorial classes of graphs Minimal superfactorial classes Are there more boundary classes? There are five more boundary classes, which can be easly obtained from the class of forests. Two of them are: 1 complements of forests; 2 bipartite complements of forests; 1 5 2 6 3 7 4 8 F 1 5 2 6 3 7 4 8 Bipartite complement of F Question Are there other boundary classes? 23 / 28
  60. 60. Boundary properties of factorial classes of graphs Minimal superfactorial classes 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 of the following three classes: X ∩ B, X ∩ B и X ∩ S is factorial and each of these classes is at most factorial. B – bipartite graphs B – complements of bipartite graphs S – split graphs 24 / 28
  61. 61. Boundary properties of factorial classes of graphs Minimal superfactorial classes Proper boundary subclasses B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .) 25 / 28
  62. 62. Boundary properties of factorial classes of graphs Minimal superfactorial classes Proper boundary subclasses B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .) superfactorial superfactorial 25 / 28
  63. 63. Boundary properties of factorial classes of graphs Minimal superfactorial classes Proper boundary subclasses B2 = Free(C4) ∩ B CB = Free(C3, C5, C6, . . .) superfactorial superfactorial i≥1 Bi = F ⊂ B2 i≥1 Bi = F ⊂ CB 25 / 28
  64. 64. Boundary properties of factorial classes of graphs Minimal superfactorial classes 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 25 / 28
  65. 65. Boundary properties of factorial classes of graphs Minimal superfactorial classes 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 subclass contained in X. We say that S is a proper boundary subclass of X if there is a sequence of superfactorial subclasses of X converging to S. 25 / 28
  66. 66. Boundary properties of factorial classes of graphs Minimal superfactorial classes Proper boundary subclasses Theorem There are no proper boundary subclasses of chordal bipartite graphs. 26 / 28
  67. 67. Boundary properties of factorial classes of graphs Minimal superfactorial classes Proper boundary subclasses Theorem There are no proper boundary subclasses of chordal bipartite graphs. Theorem The class of forests is the only proper boundary subclass of B2. 26 / 28
  68. 68. Boundary properties of factorial classes of graphs Minimal superfactorial classes Open problems Open question Find a minimal superfactorial class. 27 / 28
  69. 69. Boundary properties of factorial classes of graphs Minimal superfactorial classes Open problems Open question Find a minimal superfactorial class. Open question Is the list of boundary classes we found complete? 27 / 28
  70. 70. Thank you!

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