  We study the complexity of the Graph Isomorphism problem on graph classes
that are characterized by a finite number of forbidden induced subgraphs,
focusing mostly on the case of two forbidden subgraphs. We show hardness
results and develop techniques for the structural analysis of such graph
classes, which applied to the case of two forbidden subgraphs give the
following results: A dichotomy into isomorphism complete and polynomial-time
solvable graph classes for all but finitely many cases, whenever neither of the
forbidden graphs is a clique, a pan, or a complement of these graphs. Further
reducing the remaining open cases we show that (with respect to graph
isomorphism) forbidding a pan is equivalent to forbidding a clique of size
three.
