Vulnerability metrics play a key role in the understanding of cascading failures and target/random attacks to a network. The graph fragmentation problem (GFP) is the result of a worst-case analysis of a random attack. We can choose a fixed number of individuals for protection, and a nonprotected target node immediately destroys all reachable nodes. The goal is to minimize the expected number of destroyed nodes in the network. In this paper, we address the GFP by several approaches: metaheuristics, approximation algorithms, polytime methods for specific instances, and exact methods for small instances. The computational complexity of the GFP is included in our analysis, where we formally prove that the corresponding decision version of the problem is N P-complete. Furthermore, a strong inapproximability result holds: there is no polynomial approximation algorithm with factor lower than 5/3, unless P = N P. This promotes the study of specific instances of the problem for tractability and/or exact methods in exponential time. As a synthesis, we propose new vulnerability/connectivity metrics and an interplay with game theory using a closely related combinatorial problem called component order connectivity.42 Aprile et al. / Intl. Trans. in Op. Res. 26 (2019) 41-53 system. More recently, the focus moved toward disaster management, centrality, and vulnerability metrics under random/targeted attacks (Thai and Pardalos, 2011;Mauthe et al., 2016;Gouveia and Leitner, 2017).Simulation tools were developed in order to capture a large framework of cascading failures in epidemic modeling (Marzo et al., 2017). However, under cascading failures, the system is more robust when the individuals are poorly communicated, in a strong contrast with modern connectivity theory. To the best of our knowledge, there is no simulation tool available for both apparently antipodal scenarios.The graph fragmentation problem (GFP) is the product of a worst-case analysis of a random attack under cascading failures, therefore it is suitable for pandemic analysis. However, in its minmax version, we recover a previous problem called component order connectivity (COC). The corresponding max-min version for COC is a suitable connectivity metric.The goal of this paper is to present a comprehensive analysis for the GFP, its relation with COC, and new feasible vulnerability/connectivity metrics as a synthesis. Both GFP and COC are formally presented in Section 2. Section 3 contains a comprehensive analysis for the GFP. This section covers several approaches for the problem in different subsections, such as Complexity (Subsection 3.1), Approximation algorithms (Subsection 3.2), Polytime methods for special graphs (Subsection 3.3), Exact analysis (Subsection 3.4), and Metaheuristics (Subsection 3.5). Each subsection is enriched with references for further reading. In Section 4, we discuss vulnerability/connectivity metrics suggested by GFP and COC, and a potential interplay with game theory. Finally, Section 5 summarizes the conclusions and trends for fut...
