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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.