Abstract
The computation of edge-connected components in directed and undirected graphs is a well studied problem that is motivated by several applications (see, e.g., [Hiroshi Nagamochi and Toshihide Ibaraki, 2008]). Let G = (V,E) be a strongly connected directed graph with m edges and n vertices. An edge e ∈ E is a strong bridge if G ⧵ e is not strongly connected. More generally, a set of edges C ⊆ E is a cut if G ⧵ C is not strongly connected. If |C| = k then we refer to C as a k-sized cut of G. Hence, a strong bridge is a 1-sized cut of G. A digraph G is k-edge-connected if it has no (k-1)-cuts. We say that two vertices v and w are k-edge-connected, and we denote this relation by v ↔_{k} w, if there are k edge-disjoint directed paths from v to w and k edge-disjoint directed paths from w to v. (Note that a path from v to w and a path from w to v need not be edge-disjoint). By Menger’s theorem [Karl Menger, 1927], v ↔_{k} w if and only if the removal of any set of at most k-1 edges leaves v and w in the same strongly connected component. We define a k-edge-connected component of a digraph G = (V,E) as a maximal subset U ⊆ V such that u ↔_{k} v for all u, v ∈ U. The k-edge-connected components of G form a partition of V, since v ↔_{k} w is an equivalence relation [Loukas Georgiadis et al., 2016].
Connectivity-related problems are known to be much more difficult in directed graphs than in undirected graphs (see, e.g., [Harold N. Gabow, 2016; Monika Henzinger et al., 2020; Ken-Ichi Kawarabayashi and Mikkel Thorup, 2018]). Indeed, there is a fundamental difference in the structure of the cuts in the two scenarios. Specifically, it has been established more than 60 years ago [Gomory and Hu, 1961] that edge cuts in undirected graphs have a nice structure, as defined by the Gomory-Hu tree (or cut tree), which plays a special role in identifying, for any k, the k-edge-connected components of undirected graphs. Furthermore, many efficient algorithms for computing Gomory-Hu trees are available (see e.g., [Amir Abboud et al., 2021; Amir Abboud et al., 2022; Amir Abboud et al., 2023; Chen et al., 2022; Hariharan et al., 2007; Li et al., 2022]). On the contrary, in directed graphs edge cuts have a more complicated structure, and it was proved by Benczúr [Benczúr, 1995] that in this case cut trees do not even exist. It is thus not surprising that, while it is known how to compute the k-edge-connected components of undirected graphs in linear time for k ≤ 5 [Harold N. Gabow, 2000; Zvi Galil and Giuseppe F. Italiano, 1991; Loukas Georgiadis et al., 2021; John E. Hopcroft and Robert E. Tarjan, 1973; Kosinas, 2024; Wojciech Nadara et al., 2021; Hiroshi Nagamochi and Toshihide Ibaraki, 1992; Robert E. Tarjan, 1972; Yung H. Tsin, 2009], the situation is more challenging for directed graphs, where linear-time algorithms are only known for k ≤ 2 [Robert E. Tarjan, 1972; Loukas Georgiadis et al., 2020]. Also, as argued in [Loukas Georgiadis et al., 2023], there is a substantial increase in the inherent difficulty of the problem of computing k-edge-connected components in digraphs for k = 3 compared to k = 2. Indeed, for k = 2 any pair of vertices s,t that are not 2-edge-connected can be separated by only O(n) s-t min-cuts of size 1, for which we can define a total order [Giuseppe F. Italiano et al., 2012]. For k = 3, any pair of vertices s,t that are 2-edge-connected but not 3-edge-connected, can be separated by as many as O(n²) s-t min-cuts of size 2, which are also not totally ordered. This makes it difficult to explore the effect of removing each such cut of size 2 on the strong connectivity of the graph, similar to what was done for the case of k = 2 [Loukas Georgiadis et al., 2020]. Until recently, the best-known bound for computing the k-edge-connected components of a digraph, for constant k ≥ 3, was O(mn) by Nagamochi and Watanabe [Hiroshi Nagamochi and Toshimasa Watanabe, 1993]. Georgiadis et al. [Loukas Georgiadis et al., 2023] presented a randomized (Monte-Carlo) algorithm that computes the 3-edge-connected components of a digraph with m edges in Õ(m^{3/2}) time. Their algorithm involves a nontrivial extension of the framework of [Forster et al., 2020; Nanongkai et al., 2019] for deciding whether a digraph is (k+1)-edge-connected. It applies a local search procedure [Shiri Chechik et al., 2017; Forster et al., 2020] for identifying 2-in or 2-out sets, i.e., vertex sets S ⊆ V such that there are at most 2 edges from V ⧵ S to S or from S to V⧵ S. After finding such a set S, [Loukas Georgiadis et al., 2023] applies an efficient graph operation for replacing S with a gadget of small size that preserves the pairwise connectivity among the vertices of V ⧵ S. As in [Forster et al., 2020; Nanongkai et al., 2019], local search is initiated from sampled edges, but the overall scheme is more complicated to guarantee that enough 2-in sets or 2-out sets are identified that separate vertices that are not 3-edge-connected.
Recently, Georgiadis, Italiano and Kosinas [Georgiadis et al., 2024] improved significantly the bound of [Loukas Georgiadis et al., 2023] by showing how to compute the 3-edge-connected components of a digraph in linear time with a deterministic algorithm. Their algorithm differs substantially from [Loukas Georgiadis et al., 2023], as it is based on a new characterization of 2-sized cuts in digraphs, which requires new techniques and a suitable combination of the notions of 2-connectivity-light graphs [Loukas Georgiadis et al., 2023] and of maximally edge-disjoint strongly divergent spanning trees [Loukas Georgiadis and Robert E. Tarjan, 2015; Robert E. Tarjan, 1976]. In particular, Georgiadis, Italiano and Kosinas [Georgiadis et al., 2024] showed how to modify the minset-poset technique of Gabow [Harold N. Gabow, 2016], in order to find the 3-edge-connected components of a digraph with m edges in O(m) time.
In the invited talk, I will survey some of this recent work on higher connectivity on directed graphs.