R-connectivity Augmentation in Trees

Abstract

A \emph{vertex separator} of a connected graph $G$ is a set of vertices removing which will result in two or more connected components and a \emph{minimum vertex separator} is a set which contains the minimum number of such vertices, i.e., the cardinality of this set is least among all possible vertex separator sets. The cardinality of the minimum vertex separator refers to the connectivity of the graph G. A connected graph is said to be $k-connected$ if removing exactly $k$ vertices, $ k\geq 1$, from the graph, will result in two or more connected components and on removing any $(k-1)$ vertices, the graph is still connected. A \emph{connectivity augmentation} set is a set of edges which when augmented to a $k$-connected graph $G$ will increase the connectivity of $G$ by $r$, $r \geq 1$, making the graph $(k+r)$-$connected$ and a \emph{minimum connectivity augmentation} set is such a set which contains a minimum number of edges required to increase the connectivity by $r$. In this paper, we shall investigate a $r$-$connectivity$ augmentation in trees, $r \geq 2$. As part of lower bound study, we show that any minimum $r$-connectivity augmentation set in trees requires at least $ \lceil\frac{1}{2} \sum\limits_{i=1}^{r-1} (r-i) \times l_{i} \rceil $ edges, where $l_i$ is the number of vertices with degree $i$. Further, we shall present an algorithm that will augment a minimum number of edges to make a tree $(k+r)$-connected.