Minimum augmentation to tri-connect a bi-connected graph with upper bounds on vertex-degree

The 3-vertex-connectivity augmentation problem of a graph with degree constraints, 3VCA-DC, is defined as follows: ldquoGiven an undirected graph G = (V,E), and an upper bound f(v) isin Z⁺ cup {infin} on vertex-degree increase for each v isin V, find a smallest set E´ of edges such that (V,E cup E´) is 3-vertex-connected and such that vertex-degree increase of each v isin V by the addition of E´ to G is at most f(v), where Z⁺ is the set of nonnegative integers.rdquo In this paper we show that checking the existence of a feasible solution and finding an optimum solution to 3VCA-DC for any bi-connected graph G can be done in O(|V| + |E|) time.