1. tree -decomposition
＊ 竹内　和樹
＊ 藤井　勲

2. contents 1,Introduction 2,Main Algorithm 3,yfiles
4,Construction of Tree-Decomposition
5,Analysis of Algorithm
6,From now on Target

3. Introduction ~Purpose~ INPUT : G = (V,E)
Output : Tree-decomposition ( T , X ) １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

4. Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I }
(1), ∪ Xi = V
i∈I
(2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi
and w ∈ Xi,
(3), for all i,j,k ∈ I if j is on the path from i to k in T ,
then Xi ∩Xk ⊆ Xj

5. Introduction
(1), ∪ Xi = V
i∈I １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

6. Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I }
(1), ∪ Xi = V
i∈I
(2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi
and w ∈ Xi,
(3), for all i,j,k ∈ I if j is on the path from i to k in T ,
then Xi ∩Xk ⊆ Xj

7. Introduction
(2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi
and w ∈ Xi, １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

8. Introduction
(2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi
and w ∈ Xi, １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

9. Introduction
(2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi
and w ∈ Xi, １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

10. Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I }
(1), ∪ Xi = V
i∈I
(2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi
and w ∈ Xi,
(3), for all i,j,k ∈ I if j is on the path from i to k in T ,
then Xi ∩Xk ⊆ Xj

11. Introduction
(3), for all i,j,k ∈ I if j is on the path from i to k in T ,
then Xi ∩Xk ⊆ Xj １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

12. Introduction
(3), for all i,j,k ∈ I if j is on the path from i to k in T ,
then Xi ∩Xk ⊆ Xj １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2,4,7
4,6,7 ５

13. Introduction
(3), for all i,j,k ∈ I if j is on the path from i to k in T ,
then Xi ∩Xk ⊆ Xj １ ２ ６
1,2,3
2,3,4
4,6,8 ７ ８ ３ ４
4,5
2, 4,7
4,6,7 ５

14. Main Algorithm ～ An st-separating set ～
a set S ⊆ V {s,t} with the property that any path from s to t 1 2

15. Main Algorithm ～ Minimum st-separating set ～
The minimum number of vertices in an st-separating set

16. Calculate minimal separating
Main Algorithm ~Outline~
Input
Graph G = ( V , E )
Construct( T , X )
With | I | = 1 and X = V 1 no
∃i∈ I with
Gi not a clique
Output :current
Tree-decomposition
yes
Calculate minimal separating
Vertex set S for Gi:
m components Yi1,.....,Yim
Define Xi0 : = S
Xij : Yij ∪ S, for j = 1,....,m
Construct new ( T , X )

17. Main Algorithm １ ２ ６ ７ ８ ３ ４ ５
Input graph G = (V,I)

18. Main Algorithm １ ２ ６ ７ ８ ３ ４ ５ Calculate minimal separataing
vertex set S for Gi ７ ８ ３ ４ ５

19. Main Algorithm １ ２ ６ ７ ８ ３ ４ ５ Calculate minimal separataing
vertex set S for Gi ７ ８ ３ ４ ５

20. Main Algorithm １ ２ ６ ７ ８ ３ ４ ５ Calculate minimal separating
vertex set S for Gi . ７ ８ ３ ４ ５
Separate Y and Y S
=(4) 11 12 11

21. Main Algorithm １ ２ ６ ７ ８ ３ ４ ５ Calculate minimal separating
vertex set S for Gi . ７ ８ ３ ４ ５
Separate Y and Y S
=(4) 11 12 11

22. Main Algorithm １ ２ ６ ７ ８ ３ ４ Calculate minimal separating
vertex set S for X . ７ 11 ８ ３ ４

23. Main Algorithm １ ２ ６ ７ ８ ３ ４ Calculate minimal separating
vertex set S for X . ７ 11 ８ ３ ４

24. Main Algorithm １ ２ ６ ７ ８ ３ ４ Calculate minimal separating
vertex set S for X . ７ 11 ８ ３ ４
Separate Y and Y S
=(2,4) 21 22 21

25. Main Algorithm １ ２ ６ ７ ８ ３ ４ Calculate minimal separating
vertex set S for X . ７ 11 ８ ３ ４
Separate Y and Y S
=(2,4) 21 22 21

26. Main Algorithm ２ ６ ７ ８ ４ Calculate minimal separating
vertex set S for X . ７ 21 ８ ４

27. Main Algorithm ２ ６ ７ ８ ４ Calculate minimal separating
vertex set S for X . ７ 21 ８ ４

28. Main Algorithm ２ ６ ７ ８ ４ Calculate minimal separating
vertex set S for X . ７ 21 ８ ４
Separate Y and Y S
=(4,6) 31 32 31

29. Main Algorithm ２ ６ ７ ８ ４ Calculate minimal separating
vertex set S for X . ７ 21 ８ ４
Separate Y and Y S
=(4,6) 31 32 31

30. Main Algorithm ２ ６ ７ ４ Calculate minimal separating
vertex set S for X . ７ 31 ４

31. Main Algorithm ２ ６ ７ ４ Calculate minimal separating
vertex set S for X . ７ 31 ４

32. Main Algorithm ２ ６ ７ ４ Calculate minimal separating
vertex set S for X . ７ 31 ４
Separate Y and Y S
=(4,6) 41 42 41

33. Main Algorithm ２ ６ ７ ４ Calculate minimal separating
vertex set S for X . ７ 31 ４
Separate Y and Y S
=(4,6) 41 42 41

34. Main Algorithm ６ ７ ４ Calculate minimal separating vertex set S for X .41 ４

35. Main Algorithm ６ ７ ４ Calculate minimal separating vertex set S for X .41 ４ ・ ・ ・

36. Main Algorithm ６ ７ ４ Calculate minimal separating vertex set S for X .41 ４ ・ ・ ・
X is clique . 41
r = X = {4,6,7} 1 41

37. Main Algorithm ２ ７ ４ Calculate minimal separating vertex set S for X .42 ４

38. Main Algorithm ２ ７ ４ Calculate minimal separating vertex set S for Y .42 ４ ・ ・ ・

39. Main Algorithm ２ ７ ４ Calculate minimal separating vertex set S for Y .42 ４ ・ ・ ・
X is clique . 42
r = X = {2,4,7} 2 42

40. Main Algorithm ６ ８ ４ Calculate minimal separating vertex set S for X .32 ８ ４ ・ ・ ・
X is clique . 32
r = X = {4,6,8} 3 32

41. Main Algorithm １ ２ ３ ４ Calculate minimal separating
vertex set S for X . 22 ３ ４

42. Main Algorithm １ ２ ３ ４ Calculate minimal separating
vertex set S for X . 22 ３ ４

43. Main Algorithm １ ２ ３ ４ Calculate minimal separating
vertex set S for X . 22 ３ ４
Separate Y and Y S
=(2,3) 33 34 32

44. Main Algorithm １ ２ ３ ４ Calculate minimal separating
vertex set S for X . 22 ３ ４
Separate Y and Y S
=(2,3) 33 34 32

45. Main Algorithm ２ ３ ４ Calculate minimal separating vertex set S for X .33 ３ ４ ・ ・ ・
X is clique . 33
r = X = {2,3,4} 4 33

46. Main Algorithm １ ２ ３ Calculate minimal separating vertex set S for X .34 ３ ・ ・ ・
X is clique . 34
r = X = {1,2,3} 5 34

47. Main Algorithm ４ ５ Calculate minimal separating vertex set S for X . ・12 ４ ・ ・ ・ ５
X is clique . 12
r = X = {1,2,3} 6 12

48. Main Algorithm
All vertex sets are clique .
Connect r ,…, r . 1 6

49. Main Algorithm All vertex sets are clique . S = (2,3) S = (2,4)32 21
Connect r ,…, r . 1 6
1,2,3
2,3,4
4,6,8
S = (4,6) 31
4,5
2,4,7
4,6,7
S = (4)
S = (4,7) 1 41

50. Analysis of algorithm

51. From now on Target ・自分達の algorithm の検討
・yfiles を用いての tree-decomposition の 　　　　 ダイナミックな実装
・Heuristic の検討
？？動的な同型性判定？？

52. Ex） Given a graph G =(V,E), construct tree- decomposition (T,X).
1,2,4
7,8 1 2 4 7 3
1,3,4
4,6,7 5 6 8
3,4,5
4,5,6