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 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, 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?)