# ps6 - S | 1 Show that the minimum connected dominating set...

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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 ( |
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Unformatted text preview: 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. 2. Show that the connected vertex cover problem admits a PTAS in H-minor-free graphs. Recall that a vertex cover in a graph is a set of vertices Z such that every edge of the graph has at least one endpoint in Z ; it is called a connected vertex cover if G [ Z ] is connected....
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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.

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