A Novel Data Structure for Biconnectivity, Triconnectivity, and k-Tree Augmentation

For a connected graph, a subset of vertices of least size whose deletion increases the number of connected components is the vertex connectivity of the graph. A graph with vertex connectivity k is said to be k-vertex connected. Given a k-vertex connected graph G, vertex connectivity augmentation determines a smallest set of edges whose augmentation to G makes it (k + 1)-vertex connected. In this paper, we report our study of connectivity augmentation in 1-connected graphs, 2-connected graphs, and ktrees. For a graph, our data structure maintains the set of equivalence classes based on an equivalence relation on the set of leaves of an associated tree. This partition determines a set of edges to be augmented to increase the connectivity of the graph by one. Based on our data structure we present a new combinatorial analysis and an elegant proof of correctness of our linear time algorithm for optimum connectivity augmentation. While this is the first attempt on the study of k-tree augmentation, the study on other two augmentations is reported in the literature. Compared to other augmentations reported in the literature, we avoid recomputation of the associated tree by maintaining the data structure under edge additions.