Is there a polynomial-time algorithm to solve graph isomorphism for Delaunay graphs of (finite) hexagonal tessellations? Given a finite plane, I have a hexagonal tessellation of that plane with a fixed-size regular hexagon. I then compute the Delaunay graph G for the tessellation. Given such a graph G, I delete specific sets of nodes in that graph to yield multiple subgraphs of G. I need to determine if these subgraphs are isomorphic (to each other).

Does there exist a polynomial-time algorithm to do so?

I know that there is no know poly-time algorithm for solving graph isomorphism in the general case. But I am not sure if it is still the case for such specific Delaunay graphs.
I guess all those subgraphs will be planer graphs. And I think that there is efficient algorithm for isomorphism of planer graphs.
ref: Linear time algorithm for isomorphism of planar graphs by     J. E. Hopcroft
J. K. Wong
NOTE: I am not an expert and might not be making any sense.