Vertex Connectivity of Graphs: Algorithms and Bounds

131 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988.This thesis concerns several problems concerning vertex connectivity of undirected graphs and presents new bounds and algorithms for these problems.We have proved that the upper bound of the number of separating triplets of a triconnected graph is ${(n - 1)(n - 4)\over 2}$, and it exactly matches the lower bound, which is achieved by the wheel graph. This result has been generalized to an $O(2\sp{k}{n\sp2 \over k})$ upper bound on the number of separating k-sets in a k-connected graph. We have also obtained a new $\Omega(2\sp{k}{n\sp2 \over k\sp2})$ lower bound.Even though the upper bound for the number of separating k-sets is not linear but quadratic in n, we have obtained a linear representation for the separating k-sets of a k-connected graph. For $k = 3$ this representation is a collection of wheels, where every nonadjacent pair on the cycle of a wheel gives a separating triplet of a triconnected graph. For general k, we have obtained an $O(k\sp2 n)$ representation.We have designed a new sequential $O(n\sp2)$ algorithm for the problem of determining if the graph is four-connected or not. Consequently, we find all separating triplets of the graph if it is not four-connected. The algorithm has a parallel version which runs in O(log$\sp2 n$) time using $O(n\sp2)$ processors, which is also an improvement over $O(nm)$ processor count of the best previously known parallel algorithm.We have designed algorithms for generating all separating k-sets of a k-connected graph. The sequential algorithm runs in $O(2\sp{k}n\sp3)$ time and parallel one runs in $O(k{\rm log}n)$ deterministic parallel time or in $O({\rm log}\sp2 n)$ randomized time using $O(4\sp{k}{n\sp6 \over k\sp2})$ processors on a CRCW PRAM.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD