The Minset-Poset Approach to Representations of Graph Connectivity

Abstract

Various instances of the minimal-set poset (minset-poset for short) have been proposed in the literature, e.g., the representation of Picard and Queyranne for all st -minimum cuts of a flow network. We begin with an explanation of why this poset structure is common. We show any family of sets F that can be defined by a “labelling algorithm” (e.g., the Ford-Fulkerson labelling algorithm for maximum network flow) has an algorithm that constructs the minset poset for F . We implement this algorithm to efficiently find the nodes of the poset when F is the family of minimum edge cuts of an unweighted graph; we also give related algorithms to construct the entire poset for weighted graphs. The rest of the article discusses applications to edge- and vertex connectivity, both combinatorial and algorithmic, that we now describe. For digraphs, a natural interpretation of the minset poset represents all minimum edge cuts. In the special case of undirected graphs, the minset poset is proved to be a variant of the well-known cactus representation of all mincuts. We use the poset algorithms to construct the cactus representation for unweighted graphs in time O ( m +λ 2 n log (n/λ)) (λ is the edge connectivity) improving the previous bound O (λ n 2 ) for all but the densest graphs. We also construct the cactus representation for weighted graphs in time O ( nm log( n 2 / m )), the same bound as a previously known algorithm but in linear space O ( m ). The latter bound also holds for constructing the minset poset for any weighted digraph; the former bound also holds for constructing the nodes of that poset for any unweighted digraph. The poset is used in algorithms to increase the edge connectivity of a graph by adding the fewest edges possible. For directed and undirected graphs, weighted and unweighted, we achieve the time of the preceding two bounds, i.e., essentially the best-known bounds to compute the edge connectivity itself. Some constructions of minset posets for graph rigidity are also sketched. For vertex connectivity, the minset poset is proved to be a slight variant of the dominator tree. This leads to an algorithm to construct the dominator tree in time O ( m ) on a RAM. (The algorithm is included in the appendix, since other linear-time algorithms of similar simplicity have recently been presented.)