The problem is finding the minimum edit distance between two graphs with no edges, considering there might be different costs for adding, removing or replacing vertices.
I was told this distance is a metric, and there is a easy way to prove it. Is it so? How it can be done?
I will call the distance between two graphs G and H, d(G, H).
For any graph G, d(G, G) = 0. As long as all the costs are strictly positive, then for any G != H, d(G, H) > 0. So non-negativity is satisfied.
As long as the cost of removing a vertex is the same as the cost of adding a vertex, for any two graphs G, H, d(G, H) = d(H, G). So symmetry is satisfied.
Finally, for any three graphs G, H, K, d(G, K) is at most d(G, H) + d(H, K) because one could edit G to K by first editing it to H.
The above criteria define a metric.
