1 research outputs found
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 -vertex connected if its vertex connectivity is , . Given a -vertex connected graph , the combinatorial problem {\em
vertex connectivity augmentation} asks for a minimum number of edges whose
augmentation to makes the resulting graph -vertex connected. In this
paper, we initiate the study of -vertex connectivity augmentation whose
objective is to find a -vertex connected graph by augmenting a minimum
number of edges to a -vertex connected graph, . We shall
investigate this question for the special case when is a tree and . 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 edges, where and
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.Comment: 10 pages, 2 figures, 3 algorithms, Presented in ICGTA 201