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Definitions and preliminaries Vertex modification of totally decomposable graphs

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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


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Informations

Published by
Reads 6
Language English

Exrait

SplitvsModular

Decomposition

(thecaseoftotallydecomposablegraphs)

March29,2007

ChristophePaul

CNRS-LIRMM-Universite´MontpellierII,France

JointworkwithE.Gioan(CNRSLIRMM)

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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

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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

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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

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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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