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# hw13 - Instance An undirected graph G = V,E and an integer...

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CSCE 750, Fall 2011, Assignment 13 November 18, 2011 NOTE: We will assume that the HAMILTONIAN CIRCUIT problem (a.k.a. the HAMILTONIAN CYCLE problem) is for directed graphs: HAMILTONIAN CIRCUIT (HC) Instance: A directed graph G Question: Is there a simple directed cycle in G containing all the vertices of G ? We abbrevate this problem as HC. The book’s version (page 1062) of this problem, HAM-CYCLE, is for undirected graphs and is not the same problem . Pages 662–664 Exercises 24.3-1, 24.3-4, 24.3-6, 24.3-7 Pages 1060–1061 Exercise 34.1-1, 34.1-6 [Hint: For Kleene star closure, use dynamic programming] Pages 1065–1066 Exercise 34.2-1, 34.2-3, 34.2-6, 34.2-8 (needs concept of co-NP) Page 1077 Exercises 34.3-2 (optional), 34.3-3, 34.3-6 Pages 1085–1086 Exercises 34.4-2, 34.4-6, 34.4-7 (optional) Pages 1100–1101 Exercises 34.5-1 (easy; reduce from HAM-CYCLE), 34.5-2, 34.5-4, 34.5-6 (hint: reduce from HC), 34.5-8 (not in text) Give a polynomial reduction from VERTEX COVER to the INDEPENDENT SET problem: INDEPENDENT SET
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Unformatted text preview: Instance: An undirected graph G = ( V,E ) and an integer 0 ≤ k ≤ | V | . Question: Is there a subset I ⊆ V of vertices such that | I | ≥ k and no two vertices in I are adjacent ( independent set )? [Hint: If C is a vertex cover for G , what can you say about V-C ? If I is an independent set for G , what can you say about V-I ?] (not in text) Give a polynomial reduction from INDEPENDENT SET to CLIQUE. [Hint: The complement of an undirected graph G = ( V,E ) is the graph G = ( V,E ) where ( u,v ) ∈ E iﬀ ( u,v ) / ∈ E , for all distinct vertices u,v ∈ V .] (not in text) (more challenging) Give a polynomial reduction from (directed) HC to HAM-CYCLE (the undirected version of the Hamiltonian path problem (page 979)). [Hint: Given an instance G of HC, build a new graph G by replacing each vertex of G with a path of length 2. Connect endpoints of diﬀerent paths together with edges in some way based on the edges of G .] 1...
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