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Work on a compact oriented surface, possibly with finitely many marked points, and fix its boundary pointwise during isotopies. Suppose an Alexander system cuts the surface into discs, each containing at most one marked point. This is the filling hypothesis needed here.

A homeomorphism preserving every member up to isotopy can be adjusted by a simultaneous isotopy to preserve its structure graph of an Alexander system. If the induced graph action fixes every vertex and every oriented edge, the remaining maps on the complementary discs are isotopic to the identity by the Alexander trick. Thus the homeomorphism is isotopic to the identity. Since there are only finitely many automorphisms of the finite oriented graph, the pointwise curve-and-arc-class stabilizer is finite.^[ref]

The strong filling condition matters for a surface with boundary. A boundary-parallel Dehn twist can fix every interior curve class while remaining nontrivial relative to the boundary. A system cutting all the way into discs detects this missing information.

A collection of essential simple closed curves fills a surface if every essential simple closed curve has positive geometric intersection number with at least one member. For such curves in minimal position of curves or arcs on a closed surface of positive genus, this is equivalent to every complementary component being a disc.

The Alexander method uses a stronger cutting condition when boundary or marked points are present: an Alexander system must cut the surface into discs with at most one marked point each. For example, a pair of pants has no essential nonperipheral simple closed curves, so the empty collection passes the curve-intersection test, but it does not cut the surface into discs. The boundary-relative version must therefore keep the stronger condition explicit.

A sufficient condition is that both collections are Alexander systems: within each collection the essential simple closed curves are pairwise nonisotopic, are in pairwise minimal position of curves or arcs, have no triple intersection points, and no three curves intersect pairwise. If ${\alpha }_i$ is isotopic to ${\beta }_i$ for every $i$, the simultaneous-isotopy lemma for Alexander systems gives an ambient isotopy $H_t$ with $H_1({\alpha }_i)={\beta }_i$ for all $i$. Successive applications of the bigon criterion prove the lemma while preserving the curves already matched.

For a filling Alexander system $\{{\alpha }_i\}$, the structure graph is the embedded graphwith vertices at all curve intersections, arc endpoints, and punctures. Its edges are the curve, arc, and boundary segments between consecutive vertices. A homeomorphism preserving the system induces a graph automorphism.