This paper introduces and studies the maximum k-plex problem, which arises in social network analysis, but can also be used in several other important application areas, including wireless networks, telecommunications, and graph-based data mining. We establish NP-completeness of the decision version of the problem on arbitrary graphs. An integer programming formulation is presented and basic polyhedral study of the problem is carried out. A branch-and-cut implementation is discussed and computational test results on the proposed benchmark instances and real-life scale-free graphs are also provided.

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