6.889: Algorithms for Planar Graphs and Beyond
October 19, 2011
Problem Set 6
This problem set is due Wednesday, October 26 at noon.
For this problem set it is important to know that the separation property is defined somewhat
differently for contractionbidimensional problems. Let (
A, B, S
) be a separation of
G
and
Z
⊆
V
(
G
) be an optimal solution to a contraction bidimensional problem Π in
G
.
Let
G
A
denote the graph obtained by
contracting
each connected component of
G
[
B
] into its
adjacent vertex of
S
with smallest index, and define
G
B
similarly. Let
Z
A
denote an optimal
solution to Π in
G
A
and
Z
B
and optimal solution in
G
B
.
We say Π has the separation
property if

Z
A
 ≤ 
Z

B

+
O
(

S

) and

Z
B
 ≤ 
Z

A

+
O
(

S

) .
1. Show that the minimum connected dominating set problem admits a PTAS in apex
minorfree graphs.
A dominating set in a graph
G
is a set
D
⊆
V
(
G
) such that
D
∪
N
(
D
) =
V
(
G
), where
N
(
D
) is the set of all vertices that are neighbors of some vertex of
D
. It is called a
connected dominating set if
G
[
D
] is connected.
Solution:
First, note that upon contracting an edge, a connected dominating set
(CDS) remains connected and dominating, and hence the size of a minimum CDS does
not increase when contracting edges. Furthermore, the size of a CDS on the graph Γ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 ErikDemaine
 Algorithms, Graph Theory, vertex cover, independent set, Computational problems in graph theory, Dominating set

Click to edit the document details