Graph augmentation problems with degree-unchangeable vertices

The k-vertex-connectivity augmentation problem for a specified set of vertices of a graph with degree-unchangeable vertices, kVCA(G,S,D), is defined as follows: "Given a positive integer k, an undirected graph G = (V: E), a specified set of vertices S subset of or equal to V and a set of degree-changeable vertices D subset of or equal to V, find a smallest set of edges E' such that the vertex-connectivity of S in (V, E boolean OR E') is at least I; and E' subset of or equal to {(u,v) / u,v is an element of D}." The main result of the paper is that checking the existence of a solution and finding a solution to 2VCA(G, S, D) or 3VCA(G, S, D) can be done in O(/V/ + /E/) or O(/V/(/V/ + /E/)) time, respectively.