Smallest Bipartite Bridge-Connectivity Augmentation

Abstract

This paper addresses two augmentation problems related to bipartite graphs. The first, a fundamental graph-theoretical problem, 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, and still bipartite. The second problem, which arises naturally from research on the security of statistical data, is how to add edges so that the resulting graph is simple and does not contain any bridges. In both cases, after adding edges, the graph can be either a simple graph or, if necessary, a multi-graph. Our approach then determines whether or not such an augmentation is possible. We propose a number of simple linear-time algorithms to solve both problems. Given the well-known bridge-block data structure for an input graph, the algorithms 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. We note that there is already a polynomial time algorithm that solves the first augmentation problem related to graphs with a given general partition constraint in O(n(m+nlog n)log n) time, where m is the number of distinct edges in the input graph. We are unaware of any results for the second problem.