Definitions and preliminaries Vertex modification of totally decomposable graphs

Description

Definitions and preliminaries Vertex modification of totally decomposable graphs Conclusion and on-going work Split vs Modular Decomposition (the case of totally decomposable graphs) Christophe Paul CNRS - LIRMM - Universite Montpellier II, France March 29, 2007 Joint work with E. Gioan (CNRS LIRMM) C. Paul Split vs Modular Decomposition (the case of totally decomposable graphs)

• tree-edges between


• graphs such


• decomposable graphs


• totally decomposable


• rooted graph


• graph-labelled tree

Subjects

• Documents
• Education
• Rooted graph

Informations

• Published by: pefav
• Reads: 6
• Language: English

Exrait

SplitvsModular

Decomposition

(thecaseoftotallydecomposablegraphs)

March29,2007

ChristophePaul

CNRS-LIRMM-Universite´MontpellierII,France

JointworkwithE.Gioan(CNRSLIRMM)

)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.Ckrowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieD
edyllatotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieDConclusionandon-goingwork

3

Vertexmodificationoftotallydecomposablegraphs

2

1

Definitionsandpreliminaries

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krowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotfonoitacfithereisabijection
ρ
v
fromthetree-edgesincidentto
v
tothe
verticesof
G
v

ieachnode
v
on
k
vertices

dF∈v

oA
graph-labelledtree
isapair(
T
,
F
)with
T
atreeand
F
asetof
graphssuchthat:

mGraphlabeledtree

xetreVseiranimilerpdnasnoitinfieD)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.CGhpargaybdellebalsiTfokeergedfo
thereisabijection
ρ
v
fromthetree-edgesbetween
v
and
itschildrentotheverticesof
G
v

eachnode
v
with
k
childrenof
T
islabelledbyagraph
G
v
∈F
on
k
vertices

A
rootedgraph-labelledtree
isa
and
F
asetofgraphssuchthat:

F
)with
T
arootedtree

Rootedgraphlabeledtree

)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.C,T(riapkrowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieD
krowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotxy
∈
E
(
G
S
(
T
,
F
))iff
ρ
v
(
uv
)
ρ
v
(
vw
)
∈
E
(
G
v
),
uv
,
vw
tree-edgesonthe
x
,
y
-pathin
T
and
v
=
lca
T
(
x
,
y
).

fGivenarootedgraphlabelledtree(
T
,
F
),thegraph
G
M
(
T
,
F
)
hastheleavesof
T
asverticesand

onoitacfiidomxetreVseiranimilerpdnasnoitinfieD)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.C
xy
∈
E
(
G
S
(
T
,
F
))iff
ρ
v
(
uv
)
ρ
v
(
vw
)
∈
E
(
G
v
),
uv
,
vw
tree-edgesonthe
x
,
y
-pathin
T
and
v
=
lca
T
(
x
,
y
).

Givenarootedgraphlabelledtree(
hastheleavesof
T
asverticesand

F
),thegraph
G
M
(
T
,
F
)

)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.C,Tkrowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieD
)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDralud∀
x
∈
V
\
M
,
either
M
⊆
N
(
x
)or
M
∩
N
(
x
)=
∅

oAsubsetofvertices
M
ofagraph
G
=(
V
,
E
)isa
module
iff

MModules

svtilpSluaP.Ckrowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieD
atotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieDxy
∈
E
(
G
S
(
T
,
F
))iff
ρ
v
(
uv
)
ρ
v
(
vw
)
∈
E
(
G
v
),
∀
tree-edges
uv
,
vw
onthe
x
,
y
-pathin
T

Givenagraphlabelledtree(
T
,
F
),thegraph
G
S
(
T
,
F
)hasthe
leavesof
T
asverticesand

)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.Ckrowgniog-nodnanoisulcnoCshpargelbasopmocedyll
xy
∈
E
(
G
S
(
T
,
F
))iff
ρ
v
(
uv
)
ρ
v
(
vw
)
∈
E
(
G
v
),
∀
tree-edges
uv
,
vw
onthe
x
,
y
-pathin
T

Givenagraphlabelledtree
leavesof
T
asverticesand

F
),thegraph
G
S
(
T
,
F
)hasthe

)shpargelbasopmocedyllatotfoesaceht(noitisopmoceDraludoMsvtilpSluaP.C,T(krowgniog-nodnanoisulcnoCshpargelbasopmocedyllatotfonoitacfiidomxetreVseiranimilerpdnasnoitinfieD
.C;2>for
x
∈
A
and
y
∈
B
,
xy
∈
E
iff
x
∈
N
(
B
)and
y
∈
N
(
A
).

|
A
|
>
2,
|
B
|

split
iff

Abipartition(
A
,
B
)oftheverticesofagraph
G
=(
V
,
E
)isa

ilpSt

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