Minimum $2$-vertex strongly biconnected spanning directed subgraph problem

TL;DR

This paper investigates the problem of finding a minimum edge subset in a 2-vertex-strongly biconnected directed graph to maintain its strong biconnectivity after removing any single vertex.

Contribution

It formulates the minimum 2-vertex-strongly biconnected spanning subgraph problem and provides insights into its computational complexity and solution approaches.

Findings

The problem is computationally challenging and may be NP-hard.

Characterization of 2-vertex-strongly biconnected graphs.

Potential algorithms or heuristics for finding minimum subgraphs.

Abstract

A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ is strongly biconnected for every vertex $w\in V$. In this paper we study the following problem. Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected.