On 2-strong connectivity orientations of mixed graphs and related problems
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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,…,C_k with the property that by removing any edge e from G (directed or undirected), there is an orientation R_i of G∖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 ∈ C are in the same twinless strongly connected component of G ∖ 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.
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