Affiliation:
1. University of Ioannina, Ioannina, Greece
2. Università di Roma “Tor Vergata”, Roma, Italy
3. “Sapienza” Università di Roma, Roma, Italy
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 article, 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. The main result of this article 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, when two query vertices
v
and
w
are not 2-edge-connected, we can produce in constant time a “witness” of this property by exhibiting 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 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.
Funder
University and Research
MIUR
Italian Ministry of Education
AMANDA
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference33 articles.
1. Dominators in Linear Time
2. Digraphs
3. Linear-Time Algorithms for Dominators and Other Path-Evaluation Problems
4. Corrigendum:
a new, simpler linear-time dominators algorithm
5. T. H. Cormen C. E. Leiserson and R. L. Rivest. 1991. Introduction to Algorithms. MIT Press Cambridge MA. T. H. Cormen C. E. Leiserson and R. L. Rivest. 1991. Introduction to Algorithms. MIT Press Cambridge MA.
Cited by
13 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献
1. On 2-Strong Connectivity Orientations of Mixed Graphs and Related Problems;2024
2. On 2-Strong Connectivity Orientations of Mixed Graphs and Related Problems;Lecture Notes in Computer Science;2023
3. Optimal vertex connectivity oracles;Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing;2022-06-09
4. Subcubic algorithms for Gomory–Hu tree in unweighted graphs;Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing;2021-06-15
5. 2-edge-twinless blocks;Bulletin des Sciences Mathématiques;2021-05