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6.889: Algorithms for Planar Graphs and Beyond
September 21, 2011
Problem Set 2
This problem set is due Wednesday, September 28 at noon.
1. Prove that any undirected planar graph
G
with nonnegative edge weights can be
transformed into an undirected planar graph
G
0
with maximum degree 3 such that,
•
for any
u,v
∈
V
(
G
),
d
G
(
u,v
) =
d
G
0
(
f
(
u
)
,f
(
v
)), where
f
:
V
(
G
)
→
V
(
G
0
) maps
vertices between
G
and
G
0
; and
• 
V
(
G
0
)

=
O
(

V
(
G
)

).
Solution:
We replace each node
u
of degree
d
by a cycle
C
u
on
d
nodes. Each edge
uv
is represented by
(1)
one node
c
uv
on the cycle
C
u
,
(2)
one node
c
vu
on the cycle
C
v
,
and
(3)
an edge
c
uv
c
vu
. The edge
c
uv
c
vu
is assigned the weight of
uv
. The edges on the
cycle have zero weight.
f
maps any node
u
to an arbitrary node of
C
u
. Shortestpath
distances are maintained. For each edge we introduce two nodes. The number of nodes
in
G
0
is thus
O
(

E
(
G
)

) =
O
(

V
(
G
)

).
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This note was uploaded on 01/20/2012 for the course CS 6.889 taught by Professor Erikdemaine during the Fall '11 term at MIT.
 Fall '11
 ErikDemaine
 Algorithms

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