Twinless articulation points and some related problems

TL;DR

This paper investigates twinless articulation points and bridges in directed graphs, introduces algorithms for identifying these elements, and explores the problem of maintaining twinless strong connectivity with minimal edge removal.

Contribution

It introduces new concepts of twinless articulation points and bridges, and provides algorithms for their detection and for finding minimal edge subsets to preserve twinless strong connectivity.

Findings

Characterization of twinless articulation points and bridges

Algorithms for detecting twinless articulation points and bridges

Method for computing 2-vertex-twinless connected components

Abstract

Let $G=(V,E)$ be a twinless strongly connected graph. a vertex $v\in V$ is a twinless articulation point if the subrgraph obtained from $G$ by removing the vertex $v$ is not twinless strongly connected. An edge $e\in E$ is a twinless bridge if the subgraph obtained from $G$ by deleting $e$ is not twiless strongly connected graph. In this paper we study twinless articulation points and twinless bridges. We also study the problem of finding a minimum cardinality edge subset $E_{1} \subseteq E$ such that the subgraph $(V,E_{1})$ is twinless strongly connected. Moreover, we present an algorithm for computing the $2$-vertex-twinless connected components of $G$.