We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is ffl-far from being monotone (i.e., every monotone function differs from f on more than an ffl fraction of the domain).

We prove lower bounds on the complexity of maintaining fully dynamic k-edge or k-vertex connectivity in plane graphs and in (k − 1)-vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, where b is the word size of the machine and n is the number of vertices in G. We also show an amortized lower bound of �(log n/(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.

A common technique for bounding the runtime required to solve a constraint satisfaction problem is to exploit the structure of the problem's constraint graph [Dechter, 92]. We show that a simple structure-based technique with a minimal space requirement, pseudo-tree search [Freuder & Quinn, 85], is capable of bounding runtime almost as effectively as the best exponential space-consuming schemes. Specifically, if we let n denote the number of variables in the problem, w * denote the exponent in the complexity function of the best structure-based techniques, and h denote the exponent from pseudotree search, we show h < {w * + 1) (lg(n) + 1). The result should allow reductions in the amount of real-time accessible memory required for predicting runtime when solving CSP equivalent problems. 1

. We present a parallel algorithm for finding triconnected components on a CRCW PRAM. The time complexity of our algorithm is O(log n) and the processor-time product is O((m + n) log log n) where n is the number of vertices, and m is the number of edges of the input graph. Our algorithm, like other parallel algorithms for this problem, is based on open ear decomposition but it employs a new technique, local replacement, to improve the complexity. Only the need to use the subroutines for connected components and integer sorting, for which no optimal parallel algorithm that runs in O(log n) time is known, prevents our algorithm from achieving optimality. 1. Introduction. A connected graph G = (V; E) is k-vertex connected if it has at least (k + 1) vertices and removal of any (k \Gamma 1) vertices leaves the graph connected. Designing efficient algorithms for determining the connectivity of graphs has been a subject of great interest in the last two decades. Applications of graph connect...

Consider a communication network G in which a limited number of link and/or node faults F might occur. A routing p for the network (a fixed path between each pair of nodes) must be chosen without any knowledge of which components might become faulty. Choosing a good routing corresponds to bounding the diameter of the surviving route graph R(G,p)/F, where two nonfaulty nodes are joined by an edge if there are no faults on the route between them. We prove a number of results concerning the diameter of surviving route graphs. We show that if p is a minimal length rout- ing, then the diameter of R(G,p)/F can be on the order of the number of nodes of G, even if F consists of only a single node. However, if G is the n-dimensional cube, the diameter of R(G,p)/F<3 for any minimal length routing p and any set of faults F with IFl<n. We also show that if F consists only of edges and does not disconnect G, then the diameter of R(G,p}/F is < 31Fl+l, while if F consists only of nodes and does not disconnect G, then the diameter of R(G,p}/F is < the sum of the degrees of the nodes in F, where in both cases p is an arbitrary minimal length routing. We conclude with one of the most important contributions of this paper: a list of interesting and apparently difficult open problems.

We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called ‘open ear decomposition’. A parallel implementation of the algorithm on a CRCW PRAM runs in O(log 2 n) parallel time using O(n + m) processors, where n is the number of vertices and m is the number of edges in the graph.