Abstract
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.
Giuseppe F. Italiano—Partially supported by the Italian Ministry of Education, University and Research (MIUR) under Project AMANDA (Algorithmics for MAssive and Networked DAta).
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References
Alstrup, S., Harel, D., Lauridsen, P.W., Thorup, M.: Dominators in linear time. SIAM Journal on Computing 28(6), 2117–32 (1999)
Buchsbaum, A.L., Georgiadis, L., Kaplan, H., Rogers, A., Tarjan, R.E., Westbrook, J.R.: Linear-time algorithms for dominators and other path-evaluation problems. SIAM Journal on Computing 38(4), 1533–1573 (2008)
Georgiadis, L., Italiano, G.F., Laura, L., Parotsidis, N.: 2-edge connectivity in directed graphs. In: Proc. 26th ACM-SIAM Symp. on Discrete Algorithms, pp. 1988–2005 (2015)
Georgiadis, L., Tarjan, R.E.: Dominator tree certification and independent spanning trees (2012). CoRR, abs/1210.8303
Henzinger, M., Krinninger, S., Loitzenbauer, V.: Finding 2-edge and 2-vertex strongly connected components in quadratic time. In: Proc. 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015) (2015)
Italiano, G.F., Laura, L., Santaroni, F.: Finding strong bridges and strong articulation points in linear time. Theoretical Computer Science 447, 74–84 (2012)
Jaberi, R.: Computing the \(2\)-blocks of directed graphs (2014). CoRR, abs/1407.6178
Jaberi, R.: On computing the \(2\)-vertex-connected components of directed graphs (2014) CoRR, abs/1401.6000
Menger, K.: Zur allgemeinen kurventheorie. Fund. Math. 10, 96–115 (1927)
Nagamochi, H., Watanabe, T.: Computing \(k\)-edge-connected components of a multigraph. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E76A(4), 513–517 (1993)
Reif, J.H., Spirakis, P.G.: Strong \(k\)-connectivity in digraphs and random digraphs. Technical Report TR-25-81, Harvard University (1981)
Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)
Tarjan, R.E.: Finding dominators in directed graphs. SIAM Journal on Computing 3(1), 62–89 (1974)
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Georgiadis, L., Italiano, G.F., Laura, L., Parotsidis, N. (2015). 2-Vertex Connectivity in Directed Graphs. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_49
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DOI: https://doi.org/10.1007/978-3-662-47672-7_49
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