Graph connectivity and its augmentation: applications of MA orderings
Discrete Applied Mathematics
A Note on n-Critical Bipartite Graphs and Its Application
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
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This paper solves the problem of making a bipartite digraph strongly connected by adding the smallest number of new edges that preserve bipartiteness. A result of Baglivo and Graver shows that this corresponds to making a two-dimensional square grid framework with cables rigid by adding the smallest number of new cables. We prove a min-max formula for the smallest number of new edges in the digraph problem and give a corresponding linear-time algorithm. We generalize these results to the problem of making an arbitrary digraph strongly connected by adding the smallest number of new edges, each of which joins vertices in distinct blocks of a given partition of the vertex set.