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In the Vertex Triangle 2-Club problem, we are given an undirected graph $G$and aim to find a maximum-vertex subgraph of $G$ that has diameter at most 2and in which every vertex is contained in at least $\ell$ triangles in thesubgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relieson an ILP formulation [Almeida and Br\'as, Comput. Oper. Res. 2019]. In thiswork, we develop a combinatorial branch-and-bound algorithm that, coupled witha set of data reduction rules, outperforms the existing implementation and isable to find optimal solutions on sparse real-world graphs with more than100,000 vertices in a few minutes. We also extend our algorithm to the EdgeTriangle 2-Club problem where the triangle constraint is imposed on all edgesof the subgraph.

Efficient Branch-and-Bound Algorithms for Finding Triangle-Constrained 2-Clubs