A linear time algorithm for bi-connectivity augmentation of graphs with upper bounds on vertex-degree increase

The 2-vertex-connectivity augmentation problem of a graph with degree constraints, 2VCA-DC, is defined as follows: "Given an undirected graph G = (V, E) and an upper bound a(v; G) is an element of Z(+) boolean OR {infinity} on vertex-degree increase for each v is an element of V, find a smallest set E' of edges such that (V, E boolean OR E') has at least two internally-disjoint paths between any pair of vertices in V and such that vertex-degree increase of each v is an element of V by the addition of E' to G is at most a(v; G), where Z(+) is the set of nonnegative integers." In this paper we show that checking the existence of a feasible solution and finding an optimum solution to 2VCA-DC can be done in O(vertical bar V vertical bar + vertical bar E vertical bar) time.