The minimum stabbing triangulation of a set of points in the plane (mstr) was previously investigated in the literature. The complexity of the mstr remains open and, to our knowledge, no exact algorithm was proposed and no computational results were reported earlier in the literature of the problem. This paper presents integer programming (ip) formulations for the mstr, that allow us to solve it exactly through ip branch-and-bound (b&b) algorithms. Moreover, one of these models is the basis for the development of a sophisticated Lagrangian heuristic for the problem. Computational tests were conducted with two instance classes comparing the performance of the latter algorithm against that of a standard (exact) b&b. The results reveal that the Lagrangian algorithm yielded solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times.