The graphs G1 and G2 are isomorphic if there exists a 1-1 function f mapping the verticies of G1 onto the vertices of G2, such that the edge v1→v2 exists in G1 iff the corresponding edge f(v1)→f(v2) exists in G2. In other words, when the points of G1 are properly relabeled, the resulting graph looks just like G2. Determining graph isomorphism may be np complete.
Ulam (biography) has proposed the following conjecture. If two graphs G1 and G2 have v vertices (v > 2), and the subgraphs G1-vi can be placed in 1-1 correspondence with the subgraphs G2-f(vi), such that corresponding subgraphs are isomorphic, then G1 and G2 are isomorphic. In other words, the subgraphs determine the graph. This breaks down for v = 2, since a single edge has the same subgraphs as two disconnected points.