Why should biconnected components be identified first

Most graph optimization problems are solved on each connected component of the graph separately. This requires the identification of the connected components of the graph. The paper shows here that for several graph optimization problems, including the weighted vertex cover and the independent set problems, it suffices to know how to solve the problem on each biconnected component of the graph. The additional work required to give a solution on the whole graph takes a linear additive factor at most, whereas the potential savings in total running time are substantial. The same approach applies to approximation algorithms, and the approximation error bound is at most the maximum error bound among the biconnected components.