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arXiv · 1607.07073

Incremental $2$-Edge-Connectivity in Directed Graphs

Loukas Georgiadis·Giuseppe F. Italiano·Nikos Parotsidis
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Abstract

In this paper, we initiate the study of the dynamic maintenance of $2$-edge-connectivity relationships in directed graphs. We present an algorithm that can update the $2$-edge-connected blocks of a directed graph with $n$ vertices through a sequence of $m$ edge insertions in a total of $O(mn)$ time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices $v$ and $w$ are $2$-edge-connected. Moreover, if $v$ 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 $v$. (ii) Report in $O(n)$ time all the $2$-edge-connected blocks of $G$. To the best of our knowledge, this is the first dynamic algorithm for $2$-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.

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Loukas Georgiadis, Giuseppe F. Italiano, Nikos Parotsidis. 2016-07-24. Incremental $2$-Edge-Connectivity in Directed Graphs. https://arxiv.org/abs/1607.07073

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