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Generation of Minimal Separating Sets of a Graph
Jiro HAYAKAWA Shuji TSUKIYAMA Hiromu ARIYOSHI
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Publication Date: 1999/05/25
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
separating set, cutsets, mixed cutsets, enumeration, lowest common ancestor,
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For given undirected graph G[V,E] and vertices s and t, a minimal s-t separating set denoted by Ec & Vc is a minimal set of elements (edges and/or vertices) such that deletion of the elements from G breaks all the paths between s and t, where Ec and Vc are sets of edges and vertices, respectively. In this paper, we consider a problem of generating all minimal s-t separating sets, and show that the problem can be solved in O(µ(mt(n,n))) time, where m|E|, n|V|, µ is the number of minimal s-t separating sets of G, and t(p,q) is the time needed for finding q lowest common ancestors for q pairs of vertices in a rooted tree with p vertices. Since t(n,n) can be O(n), we can generate all minimal s-t separating in linear time per s-t separating set. However, the linear time algorithm for finding the lowest common ancestors is complicated, so that it is not efficient for a moderate size graph. Therefore, we use an O(nα (n))-time algorithm for finding the lowest common ancestors, and propose an algorithm to generate all minimal s-t separating sets in O(mnα(n)) time per s-t separating set, where α(n) is the pseudo-inverse of Ackermann function.