Strong subgraph k‐connectivity

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connectivity is, in a sense, harder to compute than generalized $k$-connectivity. However, strong subgraph $k$-connectivity can be computed in polynomial time for semicomplete digraphs and symmetric digraphs. We also provide sharp bounds on strong subgraph $k$-connectivity and pose some open questions

Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices

Faster Algorithms for Computing Maximal 2-Connected Subgraphs in Sparse Directed Graphs

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with $\widetilde{O}(mn)$ time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced $O(n^2)$ time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time $O(m^{3/2})$, which further improves the running times for sparse graphs. The notion of 2-connectivity naturally generalizes to k-connectivity for $k>2$. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time $O(m^{3/2} \log{n})$, improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in $O(n^2 \log n)$ time.Comment: Revised version of SODA 2017 paper including details for k-edge-connected subgraph

Subgraphs and network motifs in geometric networks

Many real-world networks describe systems in which interactions decay with the distance between nodes. Examples include systems constrained in real space such as transportation and communication networks, as well as systems constrained in abstract spaces such as multivariate biological or economic datasets and models of social networks. These networks often display network motifs: subgraphs that recur in the network much more often than in randomized networks. To understand the origin of the network motifs in these networks, it is important to study the subgraphs and network motifs that arise solely from geometric constraints. To address this, we analyze geometric network models, in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes. We present analytical solutions for the numbers of all 3 and 4-node subgraphs, in both directed and non-directed geometric networks. We also analyze geometric networks with arbitrary degree sequences, and models with a field that biases for directed edges in one direction. Scaling rules for scaling of subgraph numbers with system size, lattice dimension and interaction range are given. Several invariant measures are found, such as the ratio of feedback and feed-forward loops, which do not depend on system size, dimension or connectivity function. We find that network motifs in many real-world networks, including social networks and neuronal networks, are not captured solely by these geometric models. This is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information-processing functions.Comment: 9 pages, 6 figure

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs

Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known results. For \textsf{2EC-B} and \textsf{2EC-B-C}, we present new $4$-approximation algorithms that run in linear time. We also propose various heuristics to improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios