6.889: Algorithms for Planar Graphs and Beyond
Problem Set 4  Solutions
1.
Solution:
(a) Let (
T,
(
B
t
)
t
∈
V
(
T
)
) be a tree decomposition of
G
. We show that there exists a bag
B
z
such that
C
⊆
B
z
. For any edge
uv
of
T
, define
T
u
uv
as the connected component of
T

uv
that contains
u
. Let
W
u
be the union of all bags in
T
u
uv
and define
W
v
analogously. Now notice that at least
one of
W
u
and
W
v
must contain all the vertices of
C
: for otherwise, there were vertices
x, y
∈
C
such that
x
appears only in
W
u
and
y
appears only in
W
v
, and thus there would be no bag that
contains both of
x
and
y
– a contradiction.
If
C
⊆
W
u
, direct the edge
uv
towards
u
; otherwise direct it towards
v
. Since
T
is a tree, there
is a vertex
z
∈
V
(
T
) such that all the edges incident to
z
are directed towards
z
. We claim that,
C
⊆
B
z
. Suppose not; let
x
∈
C
be a vertex not in
B
z
. Consider a neighbor
u
of
z
such that
x
appears in the bags of
T
u
zu
. But since the edge
zu
is directed towards
z
, there must also be a bag
in
T
z
zu
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 Fall '11
 ErikDemaine
 Algorithms, Graph Theory, Planar graph, tree decomposition, u. Let Wu, radial graph

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