  We resolve the computational complexity of Graph Isomorphism for classes of
graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$ for all
but six pairs $(H_1,H_2)$. Schweitzer had previously shown that the number of
open cases was finite, but without specifying the open cases. Grohe and
Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph
classes of bounded clique-width. Our work combines known results such as these
with new results. By exploiting a relationship between Graph Isomorphism and
clique-width, we simultaneously reduce the number of open cases for boundedness
of clique-width for $(H_1,H_2)$-free graphs to five.
