contents 1 Introduction 2 Main Algorithm 3 yfiles

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

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

Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i } (1), ∪i∈I Xi = V (2), for every edge {v, w} ∈E, ∈ I 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

Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i } (1), ∪i∈I Xi = V (2), for every edge {v, w} ∈E, ∈ I 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

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

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

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

Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i } (1), ∪i∈I Xi = V (2), for every edge {v, w} ∈E, ∈ I 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

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, 4 4, 6, 8 4, 5 2, 4, 7 4, 6, 7 ７ ３ ４ ５ ８

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 4 4, 6, 8 2, 4, 7 4, 6, 7 2, 3, ７ ３ ４ ５ ８ 4, 5

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 4 4, 6, 8 4, 7 4, 6, 7 2, 3, ７ ３ ４ ５ ８ 4, 5 2,

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

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

Main Algorithm ~Outline~ Input Graph G = ( V , E ) Construct( T , X ) With | I | = 1 and X 1= V no ∃i∈ I with Output : current Tree-decomposition Gi not a clique yes Calculate minimal separating Vertex set S for Gi: m components Yi 1, . . . , Yim Define Xi 0 : = S Xij : Yij ∪ S, for j = 1, . . , m Construct new ( T , X )

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Analysis of algorithm

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