  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 2
and in which every vertex is contained in at least $\ell$ triangles in the
subgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relies
on an ILP formulation [Almeida and Br\'as, Comput. Oper. Res. 2019]. In this
work, we develop a combinatorial branch-and-bound algorithm that, coupled with
a set of data reduction rules, outperforms the existing implementation and is
able to find optimal solutions on sparse real-world graphs with more than
100,000 vertices in a few minutes. We also extend our algorithm to the Edge
Triangle 2-Club problem where the triangle constraint is imposed on all edges
of the subgraph.
