For a connected graph, a {em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a graph is $k$-vertex connected if its vertex connectivity is $k$, $kgeq 1$. Given a $k$-vertex connected graph $G$, the combinatorial problem {em vertex connectivity augmentation} asks for a minimum number of edges whose augmentation to $G$ makes the resulting graph $(k+1)$-vertex connected. In this paper, we initiate the study of $r$-vertex connectivity augmentation whose objective is to find a $(k+r)$-vertex connected graph by augmenting a minimum number of edges to a $k$-vertex connected graph, $r geq 1$. We shall investigate this question for the special case when $G$ is a tree and $r=2$. In particular, we present a polynomial-time algorithm to find a minimum set of edges whose augmentation to a tree makes it 3-vertex connected. Using lower bound arguments, we show that any tri-vertex connectivity augmentation of trees requires at least $lceil frac {2l_1+l_2}{2} rceil$ edges, where $l_1$ and $l_2$ denote the number of degree one vertices and degree two vertices, respectively. Further, we establish that our algorithm indeed augments this number, thus yielding an optimum algorithm.
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