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The structure graph of an Alexander system is the embedded graphwith vertices at the intersection points, punctures, and any proper-arc endpoints. Its edges are the curve, arc, and boundary segments between consecutive vertices.

The Alexander method says that if an Alexander system $\Gamma$ fills $S$, then a homeomorphism preserving the isotopy class of every member is determined up to isotopy by its induced structure-graph automorphism. In particular, a homeomorphism inducing the identity is isotopic to the identity, and the stabilizer of all curve classes is finite.

To prove this, use part a to isotope the homeomorphism so that it carries the entire embedded union $|\Gamma |$ to itself. Its remaining action on that union is exactly the structure-graph automorphism. If this action is trivial, another isotopy fixes $|\Gamma |$ pointwise. Since $\Gamma$ fills, every component of $S\setminus |\Gamma |$ is a disc. The restriction to each closed complementary disc fixes its boundary, so the Alexander trick isotopes it to the identity there. These isotopies agree on their fixed boundaries and combine into an isotopy of the whole surface. The same argument shows that two homeomorphisms with the same graph action are isotopic.