Core Concepts

Given a mixed graph G, we can efficiently compute its maximal sets of vertices C1, C2, ..., Ck such that for every i and every edge e of G, there is an orientation of G \ e where all vertices in Ci are strongly connected.

Abstract

The key insights and highlights of the content are:
The author introduces the problem of computing the "edge-resilient strongly orientable blocks" of a mixed graph G. These are the maximal sets of vertices C1, C2, ..., Ck such that for every i and every edge e of G (directed or undirected), there is an orientation of G \ e where all vertices in Ci are strongly connected.
The author shows that this problem can be reduced to the computation of the 2-edge twinless strongly connected components (2eTSCCs) of a directed graph.
The author presents a linear-time algorithm for computing the 2eTSCCs by constructing a collection of auxiliary graphs that preserve the 2eTSCCs and have a simple structure after the deletion of any edge.
The algorithm works by first splitting the directed edges and replacing the undirected edges with a special gadget in the input mixed graph G. It then computes the 2eTSCCs of the resulting directed graph, and returns the sets of ordinary vertices in these components as the edge-resilient strongly orientable blocks of G.
As a consequence of the algorithm, the author shows that the edge-resilient strongly orientable blocks form a partition of the vertex set of the input mixed graph G.

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by Loukas Georg... at **arxiv.org** 04-15-2024

Deeper Inquiries

The edge-resilient strongly orientable blocks in mixed graphs have various potential applications in different fields. One application could be in network design, where ensuring strong connectivity between specific sets of vertices is crucial. For example, in telecommunication networks, these blocks could represent groups of nodes that need to maintain strong connectivity for efficient data transmission. In road networks, these blocks could represent areas where traffic flow needs to be consistently maintained. Additionally, in structural engineering, these blocks could represent sections of a building that need to be structurally stable and interconnected. Overall, the edge-resilient strongly orientable blocks provide a way to identify and analyze specific regions within a network that require robust connectivity.

The concepts and techniques developed in this work can be extended to solve other connectivity problems in mixed graphs or directed graphs. One extension could involve exploring higher levels of connectivity beyond 2-edge resilience. By adapting the algorithms and definitions to accommodate k-edge resilience for k greater than 2, it would be possible to identify and analyze sets of vertices that remain strongly connected even after the removal of multiple edges. This extension could have applications in scenarios where maintaining connectivity under more severe disruptions is critical, such as in disaster response networks or critical infrastructure systems. Additionally, the methods developed here could be applied to study different types of connectivity constraints or properties in graphs, leading to a deeper understanding of network resilience and robustness.

Generalizing the notion of edge-resilient strong orientability to higher levels of connectivity, such as k-edge resilience for k greater than 2, would involve adapting the definitions and algorithms to handle the increased complexity. By considering the impact of removing multiple edges on the connectivity of sets of vertices, it would be possible to define and identify k-edge resilient blocks in mixed graphs. This generalization could provide insights into the robustness of networks under more severe disruptions and help in designing systems that can withstand multiple edge failures. By extending the concept to higher levels of connectivity, a more comprehensive understanding of network resilience and stability could be achieved, leading to improved network design and management strategies.

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