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Abstract The analysis of social and biological networks often involves modeling clusters of interest ascliquesor their graph‐theoretic generalizations. The ‐club model, which relaxes the requirement of pairwise adjacency in a clique to length‐bounded paths inside the cluster, has been used to model cohesive subgroups in social networks and functional modules or complexes in biological networks. However, if the graphs are time‐varying, or if they change under different conditions, we may be interested in clusters that preserve their property over time or under changes in conditions. To model such clusters that are conserved in a collection of graphs, we consider across‐graph‐clubmodel, a subset of nodes that forms a ‐club in every graph in the collection. In this article, we consider the canonical optimization problem of finding a cross‐graph ‐club of maximum cardinality in a graph collection. We develop integer programming approaches to solve this problem. Specifically, we introduce strengthened formulations, valid inequalities, and branch‐and‐cut algorithms based on delayed constraint generation. The results of our computational study indicate the significant benefits of using the approaches we introduce. 

Cliques and their generalizations are frequently used to model “tightly knit” clusters in graphs and identifying such clusters is a popular technique used in graph-based data mining. One such model is the s-club, which is a vertex subset that induces a subgraph of diameter at most s. This model has found use in a variety of fields because low-diameter clusters have practical significance in many applications. As this property is not hereditary on vertex-induced subgraphs, the diameter of a subgraph could increase upon the removal of some vertices and the subgraph could even become disconnected. For example, star graphs have diameter two but can be disconnected by removing the central vertex. The pursuit of a fault-tolerant extension of the s-club model has spawned two variants that we study in this article: robust s-clubs and hereditary s-clubs. We analyze the complexity of the verification and optimization problems associated with these variants. Then, we propose cut-like integer programming formulations for both variants whenever possible and investigate the separation complexity of the cut-like constraints. We demonstrate through our extensive computational experiments that the algorithmic ideas we introduce enable us to solve the problems to optimality on benchmark instances with several thousand vertices. This work lays the foundations for effective mathematical programming approaches for finding fault-tolerant s-clubs in large-scale networks. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Funding: The computing for this project was performed at the High Performance Computing Center at Oklahoma State University supported in part through the National Science Foundation [Grant OAC-1531128]. This material is based upon work supported by the National Science Foundation under [Grants 1662757 and 1942065]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1231 .