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

TL;DR

This paper investigates the problem of finding the smallest subset of edges in a directed graph to ensure it remains strongly biconnected even after removing any single edge, a property called 2-edge strong biconnectivity.

Contribution

It introduces the minimum 2-edge strongly biconnected spanning subgraph problem and analyzes methods to compute the smallest such subgraph.

Findings

Characterization of 2-edge strongly biconnected graphs

Algorithms for finding minimum such subgraphs

Complexity results and potential solutions

Abstract

Wu and Grumbach introduced the concept of strongly biconnected directed graphs. A directed graph $G=(V,E)$ is called strongly biconnected if the directed graph $G$ is strongly connected and the underlying undirected graph of $G$ is biconnected. A strongly biconnected directed graph $G=(V,E)$ is said to be $2$- edge strongly biconnected if it has at least three vertices and the directed subgraph $(V,E\setminus\left\lbrace e\right\rbrace )$ is strongly biconnected for all $e \in E$. Let $G=(V,E)$ be a $2$-edge-strongly biconnected directed graph. In this paper we study the problem of computing a minimum size subset $H \subseteq E$ such that the directed subgraph $(V,H)$ is $2$- edge strongly biconnected.