Permit me to include this nice image from Day & Li to illustrate @Igor's point that "in general on a surface, it [the cut locus] is a graph not a tree."

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^{The source point $p$ is on the cat's forehead, the other side in this back-view.}

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>Dey, Tamal K., and Kuiyu Li. "Cut locus and topology from surface point data." *In Proceedings of the 25th Symposium on Computational Geometry*, pp. 125-134. 2009. [ACM link](https://dl.acm.org/doi/abs/10.1145/1542362.1542390).

In the paper they compute an approximation to the cut locus on this model that pretty much follows the smooth curves drawn above.

Permit me to include this nice image from Day & Li to illustrate @Igor's point that "in general on a surface, it [the cut locus] is a graph not a tree."

---

         
^{The source point $p$ is on the cat's forehead, the other side in this rear-view.}

---

>Dey, Tamal K., and Kuiyu Li. "Cut locus and topology from surface point data." *In Proceedings of the 25th Symposium on Computational Geometry*, pp. 125-134. 2009. [ACM link](https://dl.acm.org/doi/abs/10.1145/1542362.1542390).

In the paper they compute an approximation to the cut locus on this model that pretty much follows the smooth curves drawn above.

Permit me to include this nice image from Day & Li to illustrate @Igor's point that "in general on a surface, it [the cut locus] is a graph not a tree."

---

         

---

>Dey, Tamal K., and Kuiyu Li. "Cut locus and topology from surface point data." *In Proceedings of the 25th Symposium on Computational Geometry*, pp. 125-134. 2009. [ACM link](https://dl.acm.org/doi/abs/10.1145/1542362.1542390).

In the paper they compute an approximation to the cut locus on this model that pretty much follows the smooth curves drawn above.

Permit me to include this nice image from Day & Li to illustrate @Igor's point that "in general on a surface, it [the cut locus] is a graph not a tree."

---

         
^{The source point $p$ is on the cat's forehead, the other side in this back-view.}

---

>Dey, Tamal K., and Kuiyu Li. "Cut locus and topology from surface point data." *In Proceedings of the 25th Symposium on Computational Geometry*, pp. 125-134. 2009. [ACM link](https://dl.acm.org/doi/abs/10.1145/1542362.1542390).

In the paper they compute an approximation to the cut locus on this model that pretty much follows the smooth curves drawn above.