A linear time algorithm to compute the impact of all the articulation points

TL;DR

This paper introduces a linear time algorithm to efficiently compute the impact of all articulation points in an undirected graph, significantly improving over previous methods.

Contribution

It presents the first linear time algorithm for calculating the impact of all articulation points, optimizing from a quadratic time approach.

Findings

Algorithm runs in O(m+n) time for graphs with n vertices and m edges.

Significantly faster than previous algorithms with higher complexity.

Enables efficient analysis of graph vulnerability and structure.

Abstract

The articulation points of an undirected connected graphs are those vertices whose removal increases the number of connected components of the graph, i.e. the vertices whose removal disconnects the graph. However, not all the articulation points are equal: the removal of some of them might end in a single vertex disconnected from the graph, whilst in other cases the graph can be split in several small pieces. In order to measure the effect of the removal of an articulation point, in \cite{AFL12} has been proposed the impact, defined as the number of vertices that get disconnected from the main (largest) surviving connected component (CC). In this paper we present the first linear time algorithm ($\mathcal{O}(m+n)$ for a graph with $n$ vertices and $m$ edges) to compute the impact of all the articulation points of the graph, thus improving from the $\mathcal{O}(a(m+n))\approx\mathcal{O}(nm+n^2)$ of the na\"ive algorithm, with $a$ being the number of articulation points of the graph.