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6.889: Algorithms for Planar Graphs and Beyond
October 12, 2011
Problem Set 5
This problem set is due Wednesday, October 19 at noon.
1. For
m >
2
k
, let
G
be an
m
×
m
grid, and
G
0
be the central (
m

2
k
)
×
(
m

2
k
) subgrid
of
G
. Let
N
be a set of at least
k
4
vertices in
G
0
.
(a) Show that one can designate a side of
G
as the
x
axis and the other one as the
y
axis in such a way that at least
k
2
vertices of
N
have diﬀerent
y
coordinates.
(b) Let
N
0
=
{
v
1
,...,v
k
2
} ⊆
N
be a set of exactly
k
2
vertices with diﬀerent
y

coordinates and assume that the
v
j
are sorted by increasing
y
coordinate. For
0
≤
i < k
, let
N
i
=
{
v
j
:
ki
≤
j < k
(
i
+ 1)
}
. Show (essentially by picture) that
there exist
k
disjoint paths in
G
such that each path contains exactly one vertex
out of each of
N
0
,...,N
k

1
.
(c) Show that
G
contains a model of a
k
×
k
grid
K
in which the image of each vertex
of
K
in
G
contains exactly one vertex of
N
0
.
2. Let
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 Fall '11
 ErikDemaine
 Algorithms

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