A directed graph G (V, E) is strongly connected if and only if, for any pair of vertices X and Y from V, there exists a path from X to Y and a path from Y to X. In Computer Science, the partition of a graph in strongly connected components is represented by the partition of all vertices from the graph, so that for any two vertices, X and Y, from the same partition, there exists a path from X to...

Tarjan [9], describes how depth first search can be used to identify Strongly Connected Components (SCC) of a directed graph in linear time. It is standard to study Tarjan’s SCC algorithm in most senior undergraduate or introductory graduate computer science algorithms courses. In this paper we introduce the concept of a twinless strongly connected component (TSCC) of a directed graph. Loosely ...

Tarjan’s algorihm for finding the strongly connected components of a directed graph is widely used and acclaimed. His original algorithm required at most v(2 + 5w) bits of storage, where w is the machine’s word size, whilst Nuutila and Soisalon-Soininen reduced this to v(1 + 4w). Many real world applications routinely operate on very large graphs where the storage requirements of such algorithm...

The strongly connected components of a directed graph can be found in an optimal linear time, by algorithms based on depth first search. Unfortunately, depth first search is difficult to parallelize. We describe two divide-and-conquer algorithms for this problem that have significantly greater potential for parallelization. We show the expected serial runtime of our simpler algorithm to be O(m ...

We present an Isabelle/HOL formalization of Gabow’s algorithm for finding the strongly connected components of a directed graph. Using data refinement techniques, we extract efficient code that performs comparable to a reference implementation in Java. Our style of formalization allows for re-using large parts of the proofs when defining variants of the algorithm. We demonstrate this by verifyi...

Wepropose a class of discrete-time dynamic average consensus algorithms that allow a group of agents to track the average of their reference inputs. The convergence results rely on the input-to-output stability properties of static average consensus algorithms and require that the union of communication graphs over a bounded period of time be strongly connected. The only requirement on the set ...