Determining uni-connectivity in directed graphs (original) (raw)

2-Edge Connectivity in Directed Graphs

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 2014

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.

STCON in Directed Unique-Path Graphs

We study the problem of space-efficient polynomial-time algorithms for directed st- connectivity (STCON). Given a directed graph G, and a pair of vertices s,t, the STCON problem is to decide if there exists a path from s to t in G. For general graphs, the best polynomial-time algorithm for STCON uses space that is only slightly sublinear. However, for special classes of directed graphs, polynomial-time poly-logarithmic-space algorithms are known for STCON. In this paper, we con- tinue this thread of research and study a class of graphs called unique-path graphs with respect to source s, where there is at most one simple path from s to any vertex in the graph. For these graphs, we give a polynomial-time algorithm that uses ˜ O(n#) space for any constant # ∈ (0,1). We also give a polynomial-time, ˜ O(n#)-space algorithm to recognize unique-path graphs. Unique-path graphs are related to configuration graphs of unambiguous log-space computations, but they can have some directed cycles. ...

Why Depth-First Search Efficiently Identifies Two and Three-Connected Graphs

Lecture Notes in Computer Science, 2010

Given an undirected 3-connected graph G with n vertices and m edges, we modify depth-first search to produce a sparse spanning subgraph with at most 4n − 10 edges that is still 3-connected. If G is 2-connected, to maintain 2-connectivity, the resulting graph will have at most 2n − 3 edges. The way depth-first search discards irrelevant edges illustrates the reason behind its ability to verify and certify biconnectivity [1-3] and triconnectivity in linear time. Dealing with a sparser graph, after the first depth-first-search calls, makes the algorithms in [2, 5] more efficient. We also give a characterization of separation pairs of a 2-connected graph in terms of the resulting sparse graph. *

On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

Algorithmica, 2019

For two positive integers k and , a (k ×)-spindle is the union of k pairwise internally vertexdisjoint directed paths with arcs between two vertices u and v. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed ≥ 1, finding the largest k such that an input digraph G contains a subdivision of a (k ×)-spindle is polynomialtime solvable if ≤ 3, and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results.

An Algorithm for Finding Two Node-Disjoint Paths in Arbitrary Graphs

Journal of Computing and Information Technology

Given two distinct vertices (nodes) source s and target t of a graph G = (V, E), the two node-disjoint paths problem is to identify two node-disjoint paths between s ∈ V and t ∈ V. Two paths are node-disjoint if they have no common intermediate vertices. In this paper, we present an algorithm with O(m)-time complexity for finding two node-disjoint paths between s and t in arbitrary graphs where m is the number of edges. The proposed algorithm has a wide range of applications in ensuring reliability and security of sensor, mobile and fixed communication networks.

Component Order Connectivity in Directed Graphs

Algorithmica

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A)D=(V,A)andapairofnaturalnumberskandD = ( V , A ) and a pair of natural numbers k andD=(V,A)andapairofnaturalnumberskand\ell ℓ,wearetodecidewhetherthereisasubsetXofVofsizeksuchthatthelargeststronglyconnectedcomponentinℓ , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component inℓ,wearetodecidewhetherthereisasubsetXofVofsizeksuchthatthelargeststronglyconnectedcomponentinD-XD−XhasatmostD - X has at mostD−Xhasatmost\ell ℓvertices.NotethatDCOCreducestotheDirectedFeedbackVertexSetproblemforℓ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem forℓvertices.NotethatDCOCreducestotheDirectedFeedbackVertexSetproblemfor\ell =1.ℓ=1.WestudytheparameterizedcomplexityofDCOCforgeneralandsemicompletedigraphswiththefollowingparameters:ℓ = 1 . We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters:ℓ=1.WestudytheparameterizedcomplexityofDCOCforgeneralandsemicompletedigraphswiththefollowingparameters:k, \ell ,\ell +kk,ℓ,ℓ+kandk , ℓ , ℓ + k andk,ℓ,ℓ+kandn-\ell n−ℓ.Inparticular,weprovethatDCOCwithparameterkonsemicompletedigraphscanbesolvedintimen - ℓ . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in timen−ℓ.Inparticular,weprovethatDCOCwithparameterkonsemicompletedigraphscanbesolvedintimeO^*(2^{16k})O∗(216k)butnotintimeO ∗ ( 2 16 k ) but not in timeO∗(216k)butnotintimeO^*(2^{o(k)})$$ O ∗ ( 2 o ( k ) ) unless the Exponential Time Hypothesis (ETH) fails. The upper bound ...

Extended connectivity in directed graphs

Acta chimica Slovenica, 2010

An algorithm for the evaluation of the extended connectivity in directed graphs is described and discussed. The algorithm is a general purpose one for finding the number of all paths from any given node Vi in a directed graph toward all leaves that can be reached from that particular node Vi in the graph.

On finding the strongly connected components in a directed graph

Information Processing Letters, 1994

We present two improved versions of Tarjan's algorithm for the detection of strongly connected components in a directed graph. Our new algorithms handle sparse graphs and graphs with many trivial components (containing only one node) more economically than Tarjan's original algorithm. As an application we present an e cient transitive closure algorithm.

A quick method for finding shortest pairs of disjoint paths

Networks, 1984

Let G be a directed graph containing n vertices, one of which is a distinguished source s, and m edges, each with a non-negative cost. We consider the problem of finding, for each possible sink vertex u, a pair of edge-disjoint paths from s to u of minimum total edge cost. Suurballe has given an O(n2 1ogn)-time algorithm for this problem. We give an implementation of Suurballe's algorithm that runs in O(m log(, +,+)n) time and O(m) space. Our algorithm builds an implicit representation of the n pairs of paths; given this representation, the time necessary to explicitly construct the pair of paths for any given sink is O(1) per edge on the paths.

Finding Arc and Vertex-Disjoint Paths in Networks

2009 Eighth IEEE International Conference on Dependable, Autonomic and Secure Computing, 2009

Multipath Routing plays an important role in communication networks. Multiple disjoint paths can increase the effective bandwidth between pairs of vertices, avoid congestion in a network and reduce the probability of dropped packets. In this paper, we build mathematical models for arc-disjoint paths problem and vertex-disjoint paths problem respectively, and prove that they are both Linear Programming Problem. Then we propose polynomial algorithms for finding the shortest pair of arc and vertex-disjoint paths, both with the time complexity of () Om . Furthermore, we extend these algorithms to find any k disjoint paths, whose sum-weight is minimized in time () O km .