Every edge of the Markoff graph modulo p lies on a cycle, so connected Markoff graphs are 2-connected.
The cycle structure of a Markoff automorphism over finite fields.Journal of Number Theory, 211:1–27, 2020
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Sharp vertex connectivity of the Markoff graphs modulo $p$
Every edge of the Markoff graph modulo p lies on a cycle, so connected Markoff graphs are 2-connected.