On 2-strong connectivity orientations of mixed graphs and related problems

TL;DR

This paper investigates extracting robust strongly connected groups in mixed graphs: for a mixed graph $G$, it seeks maximal sets $C_1\dots,C_k$ such that for every edge $e$, there exists an orientation of $G\setminus e$ in which all vertices in $C_i$ are strongly connected. It achieves a linear-time solution by reducing the problem to computing the $2eTSCC$ of a directed graph, where a $2eTSCC$ is a maximal subset $C$ such that any two vertices in $C$ lie in the same twinless strongly connected component of $G\setminus e$ for all edges $e$, and where a twinless SCC is a maximal strongly connected subgraph without antiparallel edges. These concepts, $TSCC$ and $2eTSCC$, enable a clean reduction and efficient implementation. Motives include road and telecommunication network design and structural stability considerations.

Abstract

A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is either $(u,v)$ or $(v,u)$. The problem of finding an orientation of a mixed graph that makes it strongly connected is well understood and can be solved in linear time. Here we introduce the following orientation problem in mixed graphs. Given a mixed graph $G$, we wish to compute its maximal sets of vertices $C_1,C_2,\ldots,C_k$ with the property that by removing any edge $e$ from $G$ (directed or undirected), there is an orientation $R_i$ of $G\setminus{e}$ such that all vertices in $C_i$ are strongly connected in $R_i$. We discuss properties of those sets, and we show how to solve this problem in linear time by reducing it to the computation of the $2$-edge twinless strongly connected components of a directed graph. A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. A $2$-edge twinless strongly connected component (2eTSCC) of $G$ is a maximal subset of vertices $C$ such that any two vertices $u, v \in C$ are in the same twinless strongly connected component of $G \setminus e$, for any edge $e$. These concepts are motivated by several diverse applications, such as the design of road and telecommunication networks, and the structural stability of buildings.