On integer programming models for the maximum 2-club problem and its robust generalizations in sparse graphs

•We propose new linear mixed-integer programs for finding maximum 2-clubs.•The new models have high-quality linear programming relaxations in sparse graphs.•We also consider a feasibility-check algorithm based on the new models.•We illustrate the effectiveness of proposed models in computational experiments.•We demonstrate how our models can be adapted to find well-connected 2-clubs.