Computing all $s$-$t$ bridges and articulation points simplified

TL;DR

This paper simplifies the process of finding all $s$-$t$ bridges and articulation points in directed graphs by reducing it to a single traversal, removing the need for complex min-cut algorithms and flow theory.

Contribution

It introduces a simplified, flow-independent method for computing all $s$-$t$ bridges and articulation points using a single graph traversal.

Findings

Simplified the algorithm for $s$-$t$ bridges and articulation points.

Reduced complexity by avoiding min-cut and flow theory.

Achieved linear-time computation with a straightforward traversal.

Abstract

Given a directed graph $G$ and a pair of nodes $s$ and $t$, an $s$-$t$ bridge of $G$ is an edge whose removal breaks all $s$-$t$ paths of $G$. Similarly, an $s$-$t$ articulation point of $G$ is a node whose removal breaks all $s$-$t$ paths of $G$. Computing the sequence of all $s$-$t$ bridges of $G$ (as well as the $s$-$t$ articulation points) is a basic graph problem, solvable in linear time using the classical min-cut algorithm. When dealing with cuts of unit size ($s$-$t$ bridges) this algorithm can be simplified to a single graph traversal from $s$ to $t$ avoiding an arbitrary $s$-$t$ path, which is interrupted at the $s$-$t$ bridges. Further, the corresponding proof is also simplified making it independent of the theory of network flows.