Generalizing and unifying earlier results of W. Mader and of the second and third authors, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edgeconnectivity prescriptions. An extension of Edmonds' theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems. 1. INTRODUCTION AND PRELIMINARIES Our main concern, the edge-connectivity augmentation problem, is as follows. What is the minimum number (or, more generally, the minimum cost) fl of new edges to be added to M so that in the resulting graph M 0 the local edge-connectivity (x; y; M 0 ) between every pair of nodes x; y is at least a prescribed value r(x; y)? Several ...

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When k is even the min-max formula for the partition-constrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge.

. We consider the problem of finding a minimum number of edges whose addition biconnects an undirected graph. This problem has been studied by several other researchers, two of whom presented a linear time algorithm for this problem in an earlier volume of this journal. However that algorithm contains an error which we expose in this paper. We present a corrected linear time algorithm for this problem as well as a new efficient parallel algorithm. The parallel algorithm runs in O(log 2 n) time using a linear number of processors on an EREW PRAM, where n is the number of vertices in the input graph. Key words. algorithm, linear time, graph augmentation, biconnected graph, parallel computation, poly-log time, EREW PRAM AMS(MOS) subject classifications. 68Q20, 68R10, 94C15, 05C40 This work was supported in part by NSF Grant CCR-89-10707. This paper appears in SIAM Journal on Computing, 1993, pp. 889-912. 1 Introduction The problem of augmenting a graph to reach a certain connectiv...

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to nd a smallest set F of new edges for which G+F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this

Given a simple graph G = (V; E), the goal is to find a smallest set F of new edges such that G = (V; E [ F ) is k-edge-connected and simple. Very recently this problem was shown to be NP-complete. In this paper we prove that if OPT k P is high enough -- depending on k only -- then OPT k S = OPT k P holds, where OPT k S (OPT k P ) is the size of an optimal solution of the augmentation problem with (without) the simplicity-preserving requirement, respectively. Furthermore, OPT k S \Gamma OPT k P g(k) holds for a certain (quadratic) function of k. Based on these facts an algorithm is given which computes an optimal solution in time O(n 4 ) for any fixed k. Some of these results are extended to the case of non-uniform demands, as well. 1 Introduction In the last decade several graph augmentation problems have been investigated. Especially the connectivity augmentation problems attracted considerable attention due to the various connections to the so-called network design ...

The minimum number of edges of an undirected graph covering a symmetric, supermodular set-function is determined. As a special case, we derive an extension of a theorem of J. Bang-Jensen and B. Jackson on hypergraph connectivity augmentation.

This paper presents a deterministic O(nm log n + n 2 log 2 n) = ~ O(nm) time algorithm for splitting o all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based on this, many graph algorithms using edge-splitting can run faster. For example, the edge-connectivity augmentation problem in an undirected multigraph can be solved in ~ O(nm) time, which is an improvement over the previously known randomized ~ O(n 3 ) bound and deterministic ~ O(n 2 m) bound. 1 Introduction Let G = (V; E) stand for an undirected multigraph with a set V of vertices and a set E of edges, where an edge with end vertices u and v is denoted by (u; v). A singleton set fxg may be simply written as x, and \ " implies proper inclusion while \ " means \ " or \ = ". For two disjoint subsets X;Y V , we denote by EG (X; Y ) the set of edges, one of whose end vertices is i...

For a given undirected graph G = (V; E; c G ) with edges weighted by nonnegative reals c G : E ! R + , let G (k) stand for the minimum amount of weights which needs to be added to make G k-edge-connected, and G 3 (k) be the resulting graph obtained from G. This paper rst shows that function G over the entire range k 2 [0; +1] can be computed in O(nm + n 2 log n) time, and then shows that all G 3 (k) in the entire range can be obtained from O(n log n) weighted cycles, and such cycles can be computed in O(nm+n 2 log n) time, where n and m are the numbers of vertices and edges, respectively. 1 Introduction Let G = (V; E; c G ) be an edge-weighted undirected graph with a set V of vertices, a set E of edges, and a weight function c G : E !R + , where R + denotes the set of nonnegative reals. We denote n = jV j and m = jEj. An edge with end vertices u and v is denoted by (u; v). A singleton set fxg may be simply written as x, and \ " implies proper inclusion while \ " ...

In the edge connectivity augmentation problem one wants to find an edge set of minimum total capacity so that if one adds it to the input graph, the maxflow value is at least k between each pair of nodes. In this paper we present the first RNC algorithm to solve this problem. The sequential version of our algorithm has running time ~O(min (n^3, tau n^2)) for undirected graphs with integer capacities, where is the amount by which the connectivity must be increased. The version not using randomization runs in time O(n^4). The most efficient earlier algorithms are those of Gabow with running times ~O(kn^2) for unit capacities and ~O(n^2 m) for general capacities, where k is the desired connectivity value.