What is an algorithm to deduce whether two colored planar graphs are isomorphic? I am aware that isomorphism is a hard problem for graphs in general, however, according to wikipedia it is possible to solve if the graphs are planar.
The application of this algorithm will be to deduce whether two planar molecules, represented with some graph-based datastructure, are the same (isomorphic). Since nodes represent atoms, the coloring of the graph is simply the type of atom (Hydrogen, Carbon, Nitrogen etc).
I claim that a node in one graph can only map through a graph isomorphism to a node in an another graph if the two nodes have the same degree.
You can create a small planar graph with a node of any desired degree by putting that node in the center, putting nodes to make up the degree around it, and creating links between the central node and all the other nodes. By shrinking this down as small as you like, you can arrange to add this, as a subgraph, to any node of a given planar graph, without making it non-planar.
Given a planar graph with coloured nodes, find the maximum degree of any node in it, and create little subgraphs of degrees above this to serve as colour markings: give each colour its own degree and link a separate little subgraph of that degree to each node of that colour.
Now solve the planar graph isomorphism on this augmented graph and you have a solution for the original graph. Similarly, any solution for the original graph can easily be turned into a solution for the augmented graph.