Efficient Algorithms to Augment the Edge-Connectivity of Specified Vertices by One in a Graph

The k-edge-connectivity augmentation problem for a specified set of vertices (kECA-SV for short) is defined by "Given a graph G = (V, E) and a subset Gamma subset of V, find a minimum set E' of edges such that G' = (V, E boolean OR E') has at least k edge-disjoint paths between any pair of vertices in Gamma." Let sigma be the edge-connectivity of Gamma (that is, G has at least sigma edge-disjoint paths between any pair of vertices in). We propose an algorithm for (sigma+1) ECA-SV which is done in O(vertical bar Gamma vertical bar) maximum flow operations. Then the time complexity is O(sigma(2)vertical bar Gamma vertical bar vertical bar V vertical bar + vertical bar E vertical bar) if a given graph is sparse, or O(vertical bar Gamma parallel to V parallel to B-G vertical bar log(vertical bar V vertical bar(2)/vertical bar B-G vertical bar) + vertical bar E vertical bar) if dense, where vertical bar B-G vertical bar is the number of pairs of adjacent vertices in G. Also mentioned is an O(vertical bar V parallel to E vertical bar + vertical bar V vertical bar(2) log vertical bar V vertical bar) time algorithm for a special case where sigma is equal to the edge-connectivity of G and an O(vertical bar V vertical bar + vertical bar E vertical bar) time one for sigma <= 2.