A new min-max theorem concerning bi-supermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph k-edge-connected. As another consequence, we solve the corresponding node-connectivity augmentation problem in directed graphs.

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimum-cost set of edges such that there are r ij vertex-disjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with non-negative costs c e 0 on all edges e 2 E. In...

In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them.

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy four-connectivity. A four-connected planar graph has no separating triangles, i.e. cycles of length 3 which are not a face. We show

Abstract not found

A k-separator (k-shredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be k-node connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of k-separators is #P-complete. However, we present an O(k )-time (deterministic) algorithm for finding all the k-shredders. This solves an open question: efficiently find a k-separator whose removal maximizes the number of connected 4, our running time is within a factor of k of the fastest algorithm known for testing k-node connectivity. One application of shredders is in increasing the node connectivity from k to (k +1)by effi tly adding an (approximately) minimum number of new edges. Jord'an [JCT(B) 1995] gaveanO(n )-time augmentation algorithm such that the number of new edges is within an additive term of (k 2) from a lower bound. We improve the running time to ), while achieving the same performance guarantee. For k 4, the running time compares favorably with the running time for testing k-node connectivity.

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

Abstract not found

The problem of connectivity augmentation consists of finding a minimum cost of new edges to be added to a given graph so as to satisfy some prescribed connectivity requirements. This paper surveys cases when polynomial time algorithms and/or good characterizations are available for the minimum. 1. INTRODUCTION In network design it is a fundamental problem to construct graphs or subgraphs of a graph of minimum cost satisfying certain connectivity specifications. Shortest paths between two specified nodes, or minimum cost spanning trees may be viewed as (well-known) special cases of this problem. Very often a starting graph is already available and the goal is to augment the graph. For example, at least how many new edges must be added to a digraph to make it strongly connected? Having such a broad class of problems (already a special case, the well-known Steiner-tree problem, has a vast literature), it is of no surprise that a large number of connectivity augmentation problems is NP...

In this paper we show that for outerplanar graphs G the problem of augmenting G by adding a mininmm nulnber of edges such that the augmented graph G is planar and bridge-connected. biconnected or triconnected can be solved in linear time and space.