Tri-connectivity Augmentation in Trees
  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 $k$, $k\geq 1$. 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. 
S. Dhanalakshmi, N. Sadagopan and D. Sunil Kumar
math.CO cs.DM