I was wondering if measuring the "edge" distance $d_e$ between two points like this had a formal name, and if it could be calculated directly? In both examples you are not allowed to cross diagonally into a cell, only traverse the edge of each cell.
For example, in the graph below $d_e=12$:
Here is another example where $d_e=6$ units:
In both cases there are multiple paths that can be taken to each point where $d_e$ is minimized (or so I think), but I'm not worried about the total possible number of paths yet, just the smallest value of $d_e$.