In this paper, we consider the augmentation problem of an undirected graph with k partitions of its vertices. The main issue is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, while maintaining the original partition constraint. To solve the problem, we propose a simple linear-time algorithm. To the best of our knowledge, the most efficient sequential algorithm runs in O(n(m+n log n) log n) time. However, we show that it can also run in O(log n) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. If a simple graph exists, our main algorithm ensures that it is as simple as possible. © 2010 Elsevier B.V. All rights reserved.
Chen, Y. C., Wei, H. W., Huang, P. C., Shih, W. K., & Hsu, T. S. (2010). The bridge-connectivity augmentation problem with a partition constraint. Theoretical Computer Science, 411(31–33), 2878–2889. https://doi.org/10.1016/j.tcs.2010.04.019