Edge connectivity between nodes and node-subsets

Let G = (V, E) be a graph where V and E are a set of nodes and a set of edges, respectively. Let X = {V-1, V-2,...V-rho} V-i subset of or equal to V be a family of node-subsets. Each node-subset V-i is called an area, and a pair of G and X is called an area graph. A node upsilon is an element of V and an area V-i is an element of X are called k-NA (node-to-area)-connected if the minimum size of a cut separating upsilon and V-i is at least k. We say that an area graph (G, X) is k-NA-edge-connected when each upsilon is an element of V and V-i is an element of X are k-NA-edge-connected. This paper gives a necessary and sufficient condition for a given (G, X) to be k-NA-edge-connected: (G, X) is k-NA-edge-connected iff, for all positive integers h less than or equal to k, every h-edge-connected component of G includes at least one node from each area or has at least k edges between the component and the rest of the nodes. This paper also studied the Minimum Area Augmentation Problem, i.e., the problem of determining whether or not a given area graph (G, X) is k-NA-edge-connected and of choosing the minimum number of nodes to be included in appropriate areas to make the area graph k-NA-edge-connected (if (G, X) is not k-NA-edge-connected). This problem can be regarded as one of the location problems, which arises from allocating service-nodes on multimedia networks. We propose an O(\E\ + \V\(2) + L' + min{\E\, k\V\}min{k\V\, k + \V\(2)}) time algorithm for solving this problem, where L' is a space required to represent output areas. For a fixed k, this algorithm also runs in linear time when the h-edge-connected components of G are available for all h = 1, 2,...,k. (C) 1998 John Wiley & Sons.