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.
Let G be a directed graph (digraph) with m edges and n vertices, and let G \ e (resp., G \ v) be the digraph obtained after deleting edge e (resp., vertex v) from G. We show how to compute in O(m + n) worst-case time:• The total number of strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G.• The size of the largest and of the smallest strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. Let G be strongly connected. We say that edge e (resp., vertex v) separates two vertices x and y, if x and y are no longer strongly connected in G \ e (resp., G \ v). We also show how to build in O(m+n) time O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs:• Report in O(n) worst-case time all the strongly connected components of G \ e (resp., G \ v), for a query edge e (resp., vertex v).• Test whether an edge or a vertex separates two query vertices in O(1) worst-case time.• Report all edges (resp., vertices) that separate two query vertices in optimal worst-case time, i.e., in time O(k), where k is the number of separating edges (resp., separating vertices). (For k = 0, the time is O(1)). All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by 1-edge and 1-vertex cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts.
Given a directed graph, two vertices v and w are 2-vertex-connected if there are two internally vertex-disjoint paths from v to w and two internally vertex-disjoint paths from w to v. In this paper, we show how to compute this relation in O(m + n) time, where n is the number of vertices and m is the number of edges of the graph. As a side result, we show how to build in linear time an O(n)-space data structure, which can answer in constant time queries on whether any two vertices are 2-vertex-connected. Additionally, when two query vertices v and w are not 2-vertex-connected, our data structure can produce in constant time a "witness" of this property, by exhibiting a vertex or an edge that is contained in all paths from v to w or in all paths from w to v. We are also able to compute in linear time a sparse certificate for 2-vertex connectivity, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-vertex connectivity properties as the input graph.
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 edge-disjoint 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. We show how to compute this relation in linear time so that we can report in constant time if two vertices are 2-edge-connected. 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.
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