final-09

# 2 we say that an undirected graph has a strongly

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2). We say that an undirected graph has a strongly connected orientation if its edges can be directed so that the resulting graph is strongly connected (meaning that for every two nodes i , j there is a directed path from i to j and from j to i ). A bridge in a graph G is an edge whose removal disconnects G . Prove: A graph G = ( N,E ) has a strongly connected orientation if and only if G is connected and has no bridge. (Ideally, your proof should be constructive. It should describe an efficient algorithm that gives a strongly connected orientation,if one exists). 3). Let G = ( N,R,B ) be a connected (multi) graph, where N are the nodes, R is a set of red edges and B is a set of blue edges. Suppose that for every node i N we have that deg R ( i ) = deg B ( i ), where deg R ( i ) is the degree of node i in G ( R ) = ( N,R ) (and deg B ( i ) is the degree of node i in G ( B ) = ( N,B )). (a). Prove that G has an Euler cycle. (b). Prove that G has an Euler cycle in which the edges of R and B alternate (i.e., when traversing the cycle, we walk along a red edge then blue then red etc.). 4). We are given a directed graph D = ( N,A ) with (nonnegative) upper bounds on the flow and nodes s,t N . An arc is upward critical if increasing the capacity on this arc (strictly) increases the max flow from s to t . An arc is

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