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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.

On $S_{0,3}$, a simple proper arc joining two distinct specified punctures is unique up to isotopy relative to its ends. Indeed, a small regular neighborhood of the arc and its two ends has one boundary curve separating those two punctures from the third; the Jordan curve theorem gives the unique such separation, and the disc it bounds gives the isotopy between any two choices.

Choose the three joining arcs, one for each puncture pair, with disjoint interiors. They form an ideal triangle graph whose complement consists of two discs. A pure homeomorphism fixes all three endpoints and carries each arc to an isotopic arc. The simultaneous-isotopy lemma makes it fix the three arcs, and the Alexander trick on each complementary disc makes it isotopic to the identity. Therefore

The pure mapping class group of the three-punctured sphere is trivial. The three simple arcs joining distinct puncture pairs form an ideal triangulation; a pure homeomorphism can be isotoped to fix these arcs, and the Alexander trick on the two complementary discs finishes the isotopy to the identity.