The bridge-connectivity augmentation problem with a partition constraint

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+nlogn)logn) time. However, we show that it can also run in O(logn) 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.