tree -decomposition ＊5499016 竹内　和樹 ＊5499023 藤井　勲

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

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

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

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

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

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

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

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

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

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

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

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

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 )

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Analysis of algorithm

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？？

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