Unformatted text preview: O ( n + m ) to decide i± the nodes o± the graph can partition into two sets N 1 and N 2 ( N = N 1 ∪ N 2 and N 1 ∩ N 2 = ∅ ) so that every red edge connects a node ±rom N 1 with a node o± N 2 , and every black edge connects two nodes within the same set (both nodes in N 1 or both nodes in N 2 ). 3). Given an undirected graph G = ( N,E ) a vertex cover C is a subset o± the nodes C ⊆ N such that ±or each edge ( i,j ) ∈ E either i ∈ C or j ∈ C or both. (a). Let L be the set o± leaves in a DFS tree o± G . (recall, a lea± is a node with no children). Prove that C = N \ L is a vertex cover. This result is used to show that one can approximate a vertex cover with a set that is at most twice the size o± the optimal size o± a set cover. (You need not show this.) (b). Now let L be the set o± leaves in a BFS tree o± G . Give an example in which C = N \ L is not a vertex cover....
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- Fall '08
- Graph Theory, vertex cover, A′, acyclic subgraph problem