Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems

The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs

Testing vertex connectivity of bowtie 1-plane graphs

A separating set of a connected graph $G$ is a set of vertices $S$ such that $G-S$ is disconnected. $S$ is a minimum separating set of $G$ if there is no separating set of $G$ with fewer vertices than $S$. The size of a minimum separating set of $G$ is called the vertex connectivity of $G$. A separating set of $G$ that is a cycle is called a separating cycle of $G$. Let $G$ be a planar graph with a given planar embedding. Let $\Lambda(G)$ be a supergraph of $G$ obtained by inserting a face vertex in each face of $G$ and connecting the face vertex to all vertices on the boundary of the face. It is well known that a set $S$ is a minimum separating set of a planar graph $G$ if and only if the vertices of $S$ can be connected together using face vertices to get a cycle $X$ of length $2|S|$ that is separating in $\Lambda(G)$. We extend this correspondence between separating sets and separating cycles from planar graphs to the class of bowtie 1-plane graphs. These are graphs that are embedded on the plane such that each edge is crossed at most once by another edge, and the endpoints of each such crossing induce either $K_4$, $K_4 \setminus \{e\}$ or $C_4$. Using this result, we give an algorithm to compute the vertex connectivity of a bowtie 1-plane graph in linear time

Directed S-T numberings, rubber bands, and testing digraph K-vertex connectivity

Directed S-T numberings, rubber bands, and testing digraph K-vertex connectivit

Directed s-t Numberings, Rubber Bands, and Testing Digraph k-Vertex Connectivity

Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k- I)-dimensional space Rk-l, ~ : V ~Rk-l, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91-102. Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algo-rithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in ex-pected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algo-rithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both al-gorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexi-ties. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of s-t numberings, we give a combinatorial construction of a directed s-t nulmberiug for any 2-vertex connected directed graph

Robust network computation

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 91-98).In this thesis, we present various models of distributed computation and algorithms for these models. The underlying theme is to come up with fast algorithms that can tolerate faults in the underlying network. We begin with the classical message-passing model of computation, surveying many known results. We give a new, universally optimal, edge-biconnectivity algorithm for the classical model. We also give a near-optimal sub-linear algorithm for identifying bridges, when all nodes are activated simultaneously. After discussing some ways in which the classical model is unrealistic, we survey known techniques for adapting the classical model to the real world. We describe a new balancing model of computation. The intent is that algorithms in this model should be automatically fault-tolerant. Existing algorithms that can be expressed in this model are discussed, including ones for clustering, maximum flow, and synchronization. We discuss the use of agents in our model, and give new agent-based algorithms for census and biconnectivity. Inspired by the balancing model, we look at two problems in more depth.(cont.) First, we give matching upper and lower bounds on the time complexity of the census algorithm, and we show how the census algorithm can be used to name nodes uniquely in a faulty network. Second, we consider using discrete harmonic functions as a computational tool. These functions are a natural exemplar of the balancing model. We prove new results concerning the stability and convergence of discrete harmonic functions, and describe a method which we call Eulerization for speeding up convergence.by David Pritchard.M.Eng