Hardness of the Minimum Steiner Tree problem on grid
This problem is $NP$-complete as shown by a reduction from Rectilinear Steiner Minimum Tree (RSMT) problem. Consider the Hanan grid: https://en.m.wikipedia.org/wiki/Hanan_grid. Since Planar-3-SAT is strongly NP-complete, we can subdivide every segment of this grid into unit segments to obtain a grid graph.