{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ps6sol

# ps6sol - 6.889 Algorithms for Planar Graphs and Beyond...

This preview shows pages 1–2. Sign up to view the full content.

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 contraction-bidimensional 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- minor-free 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.

{[ snackBarMessage ]}

### Page1 / 2

ps6sol - 6.889 Algorithms for Planar Graphs and Beyond...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online