2-Edge Connectivity in Directed Graphs

Abstract

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edgedisjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this paper is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, we also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.

Computing 2-Connected Components and Maximal 2-Connected Subgraphs in Directed Graphs: An Experimental Study

“…Indeed, two vertices may lie in different maximal 2-connected subgraphs but still be connected by several disjoint paths. This observation motivates the following natural 2-connectivity relations [9,10,14,22]. Let v and w be two distinct vertices.…”

“…Auxiliary graphs were defined in [10] and [9] to decompose the input digraph G into smaller digraphs (not necessarily subgraphs of G) that maintain, respectively, the original 2-edgeand 2-vertex-connected components of G. For 2-edgeconnectivity, we construct an auxiliary graph G Loop nesting forests. Let G s = (V, E, s) be a flow graph.…”

“…In the case of digraphs, however, only O(mn) algorithms were known (see e.g., [14,15,18,20]). It was shown only recently how to compute the 2-vertex-and 2-edge-connected components in linear time [9,10], and the best current bound for computing the maximal 2-vertex-and the 2-edge-connected subgraphs is O(min{m 3/2 , n 2 }) [5,12]. Throughout, we refer to the problems of computing the 2-vertex-and 2-edge-connected components, respectively as 2VCC and 2ECC, and to the problems of computing the maximal 2-vertex-and 2-edge-connected subgraphs, respectively as Max2VCS and Max2ECS.…”

“…To the best of our knowledge, the only previous experimental study on 2-connectivity in digraphs was carried out by Di Luigi et al [7], who compared the linear-time algorithms for computing the 2-vertex-connected components of [10] and for computing the 2-edge-connected components of [9] with simple-minded algorithms. Their experimental results show that the more sophisticated linear-time algorithms are faster than the simple-minded algorithms.…”

“…We do this by comparing the new implementations against the fastest implementations in [7] and carry out a thorough empirical analysis that highlights the merits and weaknesses of each technique. Specifically, we present an efficient implementation of a new linear-time algorithm for computing the 2-vertex and the 2-edge-connected components of G, based on loop nesting information [11], and compare it against the algorithms based on a twolevel decomposition of G using auxiliary graphs [9,10] implemented in [7]. Then, we consider the computation of the maximal 2-vertex-and 2-edgeconnected subgraphs of G, and investigate how the recent O(n 2 )-time algorithms by Henzinger et al [12], and the recent O(m 3/2 )-time algorithms by Chechik et al [5] perform in practice.…”