Problems on edges and graphs

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Reference no: EM1380153

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<b>problem:</b>
The number of strongly connected components in a graph G is k. By how much can this number change if we add a new edge?

<b>solution:</b>

Suppose if we add an edge to a biconnected graph with k strongly connected components, then there are 3-situations: the endpoints of edge lie in different strongly connected component and there is no path between 2 in the original graph, the endpoints of the edge lie in different strongly connected component and there is a path between the two in the original graph, and the endpoints of the edge lie in the same strongly connected component. In the former, the edge becomes a bridge, and thus the strongly connected components remain separate. In the middle case, the edge completes a simple cycle that includes the path between the two stongly connected components and thus the number of strongly connected components decreaseby n &#8722; 1, where n is the number of strongly connected components on that path (including the two in which the endpoints

 

Reference no: EM1380153

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