Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the *cut locus* $C$ of $\Omega$ be the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega$ for which
\begin{equation}
\lvert x - z_i \rvert = \operatorname{dist}(x,\partial \Omega).
\end{equation}

It is claimed in a paper of [Panov and Petrunin](https://arxiv.org/abs/1312.6856) that *an arbitrarily small perturbation of the boundary guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$*. (In fact, once the graph structure of $C$ is established one can show that $C$ is a tree.) They attribute this fact to [Ionin and Pestov](https://mathscinet-ams-org.libproxy.ucl.ac.uk/mathscinet-getitem?mr=107214), but unfortunately this is available only in Russian.

**Question.** How does this perturbation argument go? (And for which 'pathological' domains is it necessary in the first place?)